Abstract Algebra Exercise Solutions

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1 Groups, Orders, and Permutation Groups

Problem 1.1

Prove that every subgroup of $(ℤ, +)$ is of the form

𝑚 ℤ = {𝑚 𝑡 | 𝑡 \in ℤ},

where

𝑚 \in ℤ_{\geq 0}

Proof. Let $𝐺 \leq (ℤ, +)$ . When $𝐺 = {0}$ , $𝐺 = 0 ℤ$ . Otherwise, $𝐺$ contains at least one non-zero element and its inverse, which implies it contains at least one positive integer. Let $𝑚 = min {𝑔 \in 𝐺 | 𝑔 > 0}$ . ¹¹ Well-ordering principle.

Prove $𝐺 \subseteq 𝑚 ℤ$ . For any $𝑔 \in 𝐺$ , if $𝑔 \notin 𝑚 ℤ$ , write $𝑔 = 𝑘 𝑚 + 𝑟$ where $𝑘 \in ℤ$ , $𝑟 \in ℕ$ and $0 < 𝑟 < 𝑚$ . Then $𝑟 = 𝑔 - 𝑘 𝑚 \in 𝐺$ , contradiction.
$𝑚 ℤ \subseteq 𝐺$ is obvious because $𝑚 \in 𝐺$ and $𝑚 ℤ = ⟨ 𝑚 ⟩ \leq 𝐺$ .⁠

Problem 1.2

In a group $𝐺$ , let $𝐻_{1} \leq 𝐺$ and $𝐻_{2} \leq 𝐺$ , and define

𝐻_{1} 𝐻_{2} ≔ {ℎ_{1} ℎ_{2} | ℎ_{1} \in 𝐻_{1}, ℎ_{2} \in 𝐻_{2}} .

Give an example showing that, in general,

𝐻_{1} 𝐻_{2}

need not be a subgroup.

Proof. Let $𝐺 = 𝑆_{3}$ be a symmetric group, $𝐻_{1} = ⟨ (12) ⟩ = {𝑒, (12)}$ , $𝐻_{2} = ⟨ (23) ⟩ = {𝑒, (23)}$ .

$𝐻_{1} 𝐻_{2} = {𝑒, (12), (23), (123)}$ ²² Using the usual right-to-left composition, $(12) (23) = (123)$ ., but $(123) (12) = (13) \notin 𝐻_{1} 𝐻_{2}$ .⁠

Problem 1.3

Let

𝐺

be a finite group. Prove that for any two distinct elements

𝑎, 𝑏 \in 𝐺

𝑜 (𝑎 𝑏) = 𝑜 (𝑏 𝑎)

Note that $𝑎 𝑏$ and $𝑏 𝑎$ conjugate. In fact, conjugate elements have the same order.

Proposition 1.4

Let

𝐺

be a group. Prove that for any two conjugate elements

𝑎, 𝑏 \in 𝐺

𝑜 (𝑎 𝑏) = 𝑜 (𝑏 𝑎)

Proof. Write $𝑏 = 𝑔 𝑎 𝑔^{- 1}$ where $𝑔 \in 𝐺$ . For any positive integer $𝑛$ ,

𝑏^{𝑛} = {(𝑔 𝑎 𝑔^{- 1})}^{𝑛} = 𝑔 𝑎^{𝑛} 𝑔^{- 1},

which implies

𝑏^{𝑛} = 𝑒 \Leftrightarrow 𝑔 𝑎^{𝑛} 𝑔^{- 1} = 𝑒 \Leftrightarrow 𝑎^{𝑛} = 𝑒 .

Problem 1.5

Let $𝐺$ be a group and let $𝑔 \in 𝐺$ be an element of finite order $𝑛$ . Let $𝑟$ be a positive integer and let $𝑑 = gcd {𝑟, 𝑛}$ be the greatest common divisor of $𝑟$ and $𝑛$ . Prove that $𝑜 (𝑔^{𝑟}) = \frac{𝑛}{𝑑}$ .
Let $𝐺$ be a cyclic group of order $𝑚$ $(𝑚 > 1)$ . Determine all generators of $𝐺$ .

Proof. Note that
${(𝑔^{𝑟})}^{\frac{𝑛}{𝑑}} = {(𝑔^{𝑛})}^{\frac{𝑟}{𝑑}} = 𝑒^{\frac{𝑟}{𝑑}} = 𝑒,$
so $𝑜 (𝑔^{𝑟}) | \frac{𝑛}{𝑑}$ . There exists $𝑘 \in ℕ^{+}$ such that $\frac{𝑛}{𝑑} = 𝑘 𝑜 (𝑔^{𝑟})$ , which implies $𝑑 𝑘 | 𝑛$ .

We shall prove $𝑘 = 1$ . On the other hand,
$𝑒 = {(𝑔^{𝑟})}^{𝑜 (𝑔^{𝑟})} = 𝑔^{\frac{𝑟 𝑛}{𝑑 𝑘}} \Rightarrow 𝑛 | \frac{𝑟 𝑛}{𝑑 𝑘} \Rightarrow 𝑑 𝑘 | 𝑟 .$
So $𝑑 𝑘$ is a common divisor of $𝑟$ and $𝑛$ . Given $𝑑 = gcd {𝑟, 𝑛}$ , $𝑑 𝑘 | 𝑑$ . Hence $𝑘 = 1$ .⁠
Let $𝐺 = ⟨ 𝑔 ⟩ = {𝑒, 𝑔, 𝑔^{2}, \dots, 𝑔^{𝑚 - 1}}$ , $𝑟 \in {1, 2, \dots, 𝑚 - 1}$ . We prove the following are equivalent:
1. $𝑔^{𝑟}$ generates $𝐺$ ;
2. $𝑜 (𝑔^{𝑟}) = 𝑚$ ;
3. $gcd {𝑟, 𝑚} = 1$ .
Proof. (1) $\Leftrightarrow$ (2) is trivial. (2) $\Leftrightarrow$ (3) is a special case of (a).⁠

Problem 1.6

Let

𝐺

be a group of order

10

. Prove that

𝐺

contains at most

4

nontrivial elements of order

5

Proof. If there is no element of order $5$ , we are done. Otherwise choose $𝑔 \in 𝐺$ of order $5$ . By Problem 1.5, $𝑔^{2}, 𝑔^{3}, 𝑔^{4}$ are order $5$ elements since their exponents are coprime to $5$ . The order of $𝑔$ ensures that $𝑔, 𝑔^{2}, 𝑔^{3}, 𝑔^{4}$ are distinct.

If there is another element $ℎ \in 𝐺 \ ⟨ 𝑔 ⟩$ of order $5$ , then $ℎ, ℎ^{2}, ℎ^{3}, ℎ^{4}$ also are distinct elements of order 5. Observe that ${𝑔, 𝑔^{2}, 𝑔^{3}, 𝑔^{4}} \cap {ℎ, ℎ^{2}, ℎ^{3}, ℎ^{4}} = \emptyset$ . ³³ Otherwise one could deduce $ℎ \in {𝑔, 𝑔^{2}, 𝑔^{3}, 𝑔^{4}}$ .

Now we have identified $9$ distinct elements in $𝐺$ : ${𝑒, 𝑔, \dots, 𝑔^{4}, ℎ, \dots, ℎ^{4}}$ . Let $𝑥$ denote the remaining one. We wish to derive a contradiction by questioning its order. Recall $𝑜 (𝑥) | 𝑜 (𝐺) = 10$ .

$𝑜 (𝑥) = 1$ is definitely impossible.
$𝑜 (𝑥) = 2$ , in which case $𝑔 ℎ = ℎ 𝑔 = 𝑥$ . ⁴⁴ Verified through elimination: it always leads to contradiction if $𝑔 ℎ$ is equal to any other element in $𝐺$ . Same for $ℎ 𝑔$ . It follows that
$𝑒 = 𝑔^{5} ℎ^{5} = {(𝑔 ℎ)}^{5} = 𝑥^{5} = 𝑥,$ contradicting the choice of $𝑥$ .
$𝑜 (𝑥) = 5$ makes $𝐺$ contain $𝑥^{2}, 𝑥^{3}, 𝑥^{4}$ that are disjoint from $⟨ 𝑔 ⟩$ and $⟨ ℎ ⟩$ . So you get at least $1 + 4 + 4 + 4 = 13$ elements.
$𝑜 (𝑥) = 10$ . In $𝐺 = ⟨ 𝑥 ⟩$ , the four elements $𝑥^{2}, 𝑥^{4}, 𝑥^{6}, 𝑥^{8}$ form the unique subgroup of order $5$ . ⁵⁵ By Problem 1.5. Thus if $𝑔$ is one of them, then $ℎ$ must also lie in $⟨ 𝑔 ⟩$ , contradicting the choice of $ℎ$ .

Therefore, all order- $5$ elements lie in $⟨ 𝑔 ⟩$ , which has exactly four nontrivial elements.⁠

Problem 1.7

Let $ℝ$ denote the set of real numbers and let $ℝ^{*} = ℝ \ {0}$ . For any $𝑎 \in ℝ^{*}$ and $𝑏 \in ℝ$ , define a map $𝑓_{𝑎, 𝑏} : ℝ \to ℝ$ as follows:

𝑓_{𝑎, 𝑏} (𝑥) = 𝑎 𝑥 + 𝑏, \forall 𝑥 \in ℝ .

Let $𝐺 = {𝑓_{𝑎, 𝑏} | 𝑎 \in ℝ^{*}, 𝑏 \in ℝ}$ be the set of all such maps.

Prove that $𝐺$ forms a group under composition of maps.
Find the center of $𝐺$ .

Proof. Composition of maps is associative. For every $𝑓_{𝑎, 𝑏}, 𝑓_{𝑐, 𝑑} \in 𝐺$ :
$(𝑓_{𝑎, 𝑏} \circ 𝑓_{𝑐, 𝑑}) (𝑥) = 𝑎 (𝑐 𝑥 + 𝑑) + 𝑏 = 𝑎 𝑐 𝑥 + 𝑎 𝑑 + 𝑏 = 𝑓_{𝑎 𝑐, 𝑎 𝑑 + 𝑏} \in 𝐺,$
so $𝐺$ is closed under multiplication. Clearly, $𝑓_{1, 0}$ is the identity of $𝐺$ , and $𝑓_{𝑎, 𝑏}^{- 1} = 𝑓_{\frac{1}{𝑎}, - \frac{𝑏}{𝑎}}$ .⁠
Solution. For every $𝑓_{𝛼, 𝛽} \in 𝑍 (𝐺)$ :
$\forall 𝑓_{𝑎, 𝑏} \in 𝐺, \forall 𝑥 \in ℝ, 𝑓_{𝛼, 𝛽} 𝑓_{𝑎, 𝑏} (𝑥) = 𝑓_{𝑎, 𝑏} 𝑓_{𝛼, 𝛽} (𝑥) .$
By definition of $𝑓_{𝑎, 𝑏}$ :
$\begin{aligned} 𝛼 (𝑎 𝑥 + 𝑏) + 𝛽 & = 𝑎 (𝛼 𝑥 + 𝛽) + 𝑏 \\ 𝛽 𝑎 + (1 - 𝛼) 𝑏 & = 𝛽, & \forall 𝑎 \in ℝ^{*}, 𝑏 \in ℝ . \end{aligned}$
Hence $𝛼 = 1, 𝛽 = 0$ . $𝐺$ has a trivial center ${𝑓_{1, 0}}$ .

Problem 1.8

Let

{SL}_{2} (ℤ) = {(\begin{matrix} 𝑎 & 𝑏 \\ 𝑐 & 𝑑 \end{matrix}) | 𝑎, 𝑏, 𝑐, 𝑑 \in ℤ, 𝑎 𝑑 - 𝑏 𝑐 = 1} .

Prove that ${SL}_{2} (ℤ)$ is generated by the matrices

𝑻 = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}), 𝑺 = (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) .

Proof. Clearly, ${SL}_{2} (ℤ)$ forms a group under matrix multiplication with identity $𝑰_{𝟐}$ . It suffices to prove that, by multiplying $𝑨 = (\begin{matrix} 𝑎 & 𝑏 \\ 𝑐 & 𝑑 \end{matrix}) \in {SL}_{2} (ℤ)$ by finitely many $𝑻, 𝑺, 𝑻^{- 1}$ , we can always reduce it to $𝑰_{2}$ .

We use the notation $𝑨 \overset{𝑷}{\to} 𝑩$ to denote $𝑷 𝑨 = 𝑩$ .

When $𝑐 \neq 0$ , write $𝑎 = 𝑘 𝑐 + 𝑟$ where $𝑘 \in ℤ, 0 \leq 𝑟 < | 𝑐 |$ . Do the following transformation

(\begin{matrix} 𝑎 & 𝑏 \\ 𝑐 & 𝑑 \end{matrix}) \overset{𝑻^{- 𝑘}}{\to} (\begin{matrix} 𝑟 & * \\ 𝑐 & * \end{matrix}) \overset{𝑺}{\to} (\begin{matrix} 𝑐 & * \\ - 𝑟 & * \end{matrix})

finitely many times to reduce $𝑐$ to $0$ . ⁶⁶ This process terminates because the absolute value of the lower-left entry strictly decreases. It is essentially the Euclidean algorithm on the first column.

When $𝑐 = 0$ , $𝑨$ has the form $(\begin{matrix} 1 & 𝑏 \\ 0 & 1 \end{matrix})$ or $(\begin{matrix} - 1 & 𝑏 \\ 0 & - 1 \end{matrix})$ since $det 𝑨 = 1$ . Then we can further reduce it to $𝑰_{2}$ :

(\begin{matrix} 1 & 𝑏 \\ 0 & 1 \end{matrix}) \overset{𝑻^{- 𝑏}}{\to} 𝑰_{2};

(\begin{matrix} - 1 & 𝑏 \\ 0 & - 1 \end{matrix}) \overset{𝑻^{𝑏}}{\to} - 𝑰_{2} \overset{𝑺^{2}}{\to} 𝑰_{2} .

⁠

Problem 1.9

Compute the cycle decomposition of

𝜎 = (1357) (2345) (567) \in 𝑆_{7}

Solution.

\begin{matrix} 1 \to 1 \to 1 \to 3 \\ 3 \to 3 \to 4 \to 4 \\ 4 \to 4 \to 5 \to 7 \\ 7 \to 5 \to 2 \to 2 \\ 2 \to 2 \to 3 \to 5 \\ 5 \to 6 \to 6 \to 6 \\ 6 \to 7 \to 7 \to 1 \end{matrix}

$(1357) (2345) (567) = (1347256)$ .

Problem 1.10

Let

𝜎 \in 𝑆_{𝑛} (𝑛 > 2)

. Write

𝜎

as a product of finitely many disjoint cycles

𝜎_{1}, 𝜎_{2}, \dots, 𝜎_{𝑘}

where

𝜎_{𝑖}

has length

𝑟_{𝑖}

. Find the order of

𝜎

Lemma 1.11

Let

𝜎_{1}, 𝜎_{2}, \dots, 𝜎_{𝑘} (𝑘 \geq 2)

be disjoint cycles in

𝑆_{𝑛}

, then

⟨ 𝜎_{1} ⟩ \cap ⟨ 𝜎_{2}, \dots, 𝜎_{𝑘} ⟩ = {𝑒}

Proof. Let

𝜎 = 𝜎_{1}^{𝑎_{1}} = \prod_{𝑖 = 2}^{𝑘} 𝜎_{𝑖}^{𝑎_{𝑖}}, 𝑎_{𝑖} \in ℤ for each 1 \leq 𝑖 \leq 𝑘 .

We shall prove $𝜎 = 𝑒$ . If $𝜎_{1} = 𝑒$ , we are done. Otherwise, take $𝑥 \in supp (𝜎_{1})$ , then $𝑥 \notin supp (𝜎_{𝑖})$ for each $2 \leq 𝑖 \leq 𝑘$ , which implies $𝜎_{𝑖}^{𝑎_{𝑖}} (𝑥) = 𝑥$ . Therefore

\begin{aligned} 𝜎_{1}^{𝑎_{1}} (𝑥) & = (\prod_{𝑖 = 2}^{𝑘} 𝜎_{𝑖}^{𝑎_{𝑖}}) (𝑥) \\ = (\prod_{𝑖 = 2}^{𝑘 - 1} 𝜎_{𝑖}^{𝑎_{𝑖}}) (𝜎_{𝑘}^{𝑎_{𝑘}} (𝑥)) \\ = (\prod_{𝑖 = 2}^{𝑘 - 1} 𝜎_{𝑖}^{𝑎_{𝑖}}) (𝑥) \\ = \dots \\ = 𝜎_{2}^{𝑎_{2}} (𝑥) = 𝑥 . \end{aligned}

Let $𝑟$ be the length of $𝜎_{1}$ , since $𝜎_{1}$ is a cycle, $𝑟 ∣ 𝑎_{1}$ . Note that $𝑜 (𝜎_{1}) = 𝑟$ , so $𝜎 = 𝜎_{1}^{𝑎_{1}} = 𝑒$ .⁠

Proposition 1.12

Let $𝜎_{1}, 𝜎_{2}, \dots, 𝜎_{𝑘}$ be disjoint cycles in $𝑆_{𝑛}$ , then

𝐺 = \prod_{𝑖 = 1}^{𝑘} ⟨ 𝜎_{𝑖} ⟩

is a direct product.

Proof. Every element of $⟨ 𝜎_{𝑖} ⟩$ commutes with every element of $⟨ 𝜎_{𝑗} ⟩$ since disjoint cycles commute. For each $𝑖$ ,

\begin{aligned} ⟨ 𝜎_{𝑖} ⟩ \cap ⟨ 𝜎_{1} ⟩ \dots ⟨ 𝜎_{𝑖 - 1} ⟩ ⟨ 𝜎_{𝑖 + 1} ⟩ \dots ⟨ 𝜎_{𝑘} ⟩ & = ⟨ 𝜎_{𝑖} ⟩ \cap ⟨ 𝜎_{1}, \dots, 𝜎_{𝑖 - 1}, 𝜎_{𝑖 + 1}, \dots, 𝜎_{𝑘} ⟩ \\ = {𝑒} \end{aligned}

by Lemma 1.11.⁠

Hence

𝑜 (𝜎) = lcm {𝑜 (𝜎_{1}), \dots, 𝑜 (𝜎_{𝑘})} = lcm {𝑟_{1}, \dots, 𝑟_{𝑘}} .

Problem 1.13

Let $𝑛 > 2$ .

Let $𝑟$ denote the transformation of $𝐷_{𝑛}$ that sends the vertex pair $(1, 2)$ to $(2, 3)$ , and let $𝑠$ denote the transformation that sends $(1, 2)$ to $(2, 1)$ . Prove that
$𝐷_{𝑛} = ⟨ 𝑟, 𝑠 | 𝑟^{𝑛} = 𝑠^{2} = 𝑟 𝑠 𝑟 𝑠 = 𝑒 ⟩ .$
Find all elements of order $2$ and all elements of order $𝑛$ in $𝐷_{𝑛}$ .

Proof. According to geometry, rotations of a regular $𝑛$ -gon are $𝑟^{𝑖} (1 \leq 𝑖 \leq 𝑛 - 1)$ and reflections are $𝑟^{𝑖} 𝑠 (0 \leq 𝑖 \leq 𝑛 - 1)$ , hence
$𝐷_{𝑛} = {𝑒, 𝑟, 𝑟^{2}, \dots, 𝑟^{𝑛 - 1}, 𝑠, 𝑟 𝑠, 𝑟^{2} 𝑠, \dots, 𝑟^{𝑛 - 1} 𝑠} .$
Obviously, $𝑟^{𝑛} = 𝑠^{2} = 𝑒$ and $𝑠 𝑟 𝑠 = 𝑟^{- 1}$ , which implies $𝑟 𝑠 𝑟 𝑠 = 𝑒$ .⁠
Solution. Elements of order $𝑛$ are rotations $𝑟^{𝑖} (1 \leq 𝑖 \leq 𝑛 - 1)$ where $gcd {𝑛, 𝑖} = 1$ . ⁷⁷ By Problem 1.5. Elements of order $2$ are all reflections: $𝑟^{𝑖} 𝑠 (0 \leq 𝑖 \leq 𝑛 - 1)$ and the $180 °$ rotation $𝑟^{\frac{𝑛}{2}}$ when $𝑛$ is even.

Problem 1.14

Find the number of permutations of order

6

𝑆_{8}

Solution.

Permutations of type $[1^{3} 2^{1} 3^{1}]$ : $\frac{8!}{3! \cdot 2 \cdot 3} = 1120$ ;
Permutations of type $[1^{1} 2^{2} 3^{1}]$ : $\frac{8!}{2^{2} \cdot 2! \cdot 3} = 1680$ ;
Permutations of type $[2^{1} 3^{2}]$ : $\frac{8!}{2 \cdot 3^{2} \cdot 2!} = 1120$ ;
Permutations of type $[1^{2} 6^{1}]$ : $\frac{8!}{2! \cdot 6} = 3360$ ;
Permutations of type $[2^{1} 6^{1}]$ : $\frac{8!}{2 \cdot 6} = 3360$ .

Hence there are $10640$ permutations of order $6$ .

Problem 1.15

Let $𝜎 \in 𝑆_{𝑛}$ . Write $𝜎$ as a product of finitely many disjoint nontrivial cycles, and let $𝑐 (𝜎)$ be the sum of the number of these cycles and the number of fixed points of $𝜎$ .

Prove that $𝜎$ is an even permutation if and only if $𝑛 - 𝑐 (𝜎)$ is even.
Let $𝜏$ be a transposition. Prove that
$𝑐 (𝜎 𝜏) = 𝑐 (𝜎) + 1 or 𝑐 (𝜎 𝜏) = 𝑐 (𝜎) - 1,$ and explain when each of the two cases above occurs.
Prove that the minimum number of transpositions needed to express $𝜎$ as a product of transpositions is $𝑛 - 𝑐 (𝜎)$ .

Proof. Write $𝜎 = 𝜎_{1} 𝜎_{2} \dots 𝜎_{𝑘}$ where $𝜎_{𝑖}$ has length $𝑟_{𝑖} \geq 2$ . By definition,
$𝑘 + (𝑛 - 𝑟_{1} - 𝑟_{2} - \dots - 𝑟_{𝑘}) = 𝑐 (𝜎) .$
So $𝜎$ is an even permutation if and only if
$(𝑟_{1} - 1) + (𝑟_{2} - 1) + \dots + (𝑟_{𝑘} - 1) = 𝑟_{1} + 𝑟_{2} + \dots + 𝑟_{𝑘} - 𝑘 = 𝑛 - 𝑐 (𝜎)$
is even.⁠
Proof. Write
$𝜎_{𝑖} = (𝑎_{𝑖 1} 𝑎_{𝑖 2} \dots 𝑎_{𝑖 𝑟_{𝑖}}), 1 \leq 𝑖 \leq 𝑘 .$
Let $𝜏 = (𝑎, 𝑏)$ . Since $𝜎_{𝑖}$ and $𝜎_{𝑗}$ are disjoint for all $𝑖 \neq 𝑗$ , there are only four cases for the relation between $𝜏$ and the cycles $𝜎_{𝑖}$ . ⁸⁸ If we allow trivial cycles (fixed points) in the product representation of $𝜎$ , there would be only two cases: either $𝜏$ splits a cycle or $𝜏$ merges two cycles.
The original problem handles nontrivial cycles and fixed points separately.

Case I: $𝜏$ is disjoint from all $𝜎_{𝑖}$ , then $𝜎 𝜏 = 𝜎_{1} \dots 𝜎_{𝑘} 𝜏$ (disjoint nontrivial cycles), so $𝜎 𝜏$ gets an additional cycle $𝜏$ while losing two fixed points $𝑎$ and $𝑏$ , thus $𝑐 (𝜎 𝜏) = 𝑐 (𝜎) - 1$ .

Case II: $𝜏$ is contained in one $𝜎_{𝑖}$ and disjoint from any other $𝜎_{𝑗} (𝑗 \neq 𝑖),$ i.e. there exists $1 \leq 𝑝 \neq 𝑞 \leq 𝑟_{𝑖}$ such that $𝑎_{𝑖 𝑝} = 𝑎$ and $𝑎_{𝑖 𝑞} = 𝑏$ .

For simplicity, we only consider $𝑝 + 1 < 𝑞 < 𝑟_{𝑖}$ . Compute
$\begin{aligned} 𝜎_{𝑖} 𝜏 & = (𝑎_{𝑖 1} 𝑎_{𝑖 2} \dots 𝑎_{𝑖 𝑟_{𝑖}}) (𝑎_{𝑖 𝑝} 𝑎_{𝑖 𝑞}) \\ = (𝑎_{𝑖 𝑝} 𝑎_{𝑖, 𝑞 + 1} 𝑎_{𝑖, 𝑞 + 2} \dots 𝑎_{𝑖 𝑟_{𝑖}} 𝑎_{𝑖 1} 𝑎_{𝑖 2} \dots 𝑎_{𝑖, 𝑝 - 1}) (𝑎_{𝑖, 𝑝 + 1} 𝑎_{𝑖, 𝑝 + 2} \dots 𝑎_{𝑖 𝑞}) . \end{aligned}$
Hence $𝜎_{𝑖} 𝜏$ can be expressed as two disjoint nontrivial cycles that are also disjoint from any other $𝜎_{𝑗} (𝑗 \neq 𝑖)$ . Since disjoint cycles commute,
$𝜎 𝜏 = 𝜎_{1} 𝜎_{2} \dots 𝜎_{𝑘} 𝜏 = 𝜎_{1} \dots 𝜎_{𝑖 - 1} (𝜎_{𝑖} 𝜏) 𝜎_{𝑖 + 1} \dots 𝜎_{𝑘},$
so $𝜎 𝜏$ gets one additional cycle with fixed points unchanged, hence $𝑐 (𝜎 𝜏) = 𝑐 (𝜎) + 1$ .

Similarly, one can compute $𝜎_{𝑖} 𝜏$ under other conditions of $𝑝$ and $𝑞$ , and $𝑐 (𝜎 𝜏) = 𝑐 (𝜎) + 1$ still holds.

Case III: one endpoint of $𝜏$ lies in $𝜎_{𝑖}$ and the other lies in $𝜎_{𝑗} (𝑖 \neq 𝑗)$ . Similarly, one can prove $𝜎 𝜏$ loses one cycle with fixed points unchanged, hence $𝑐 (𝜎 𝜏) = 𝑐 (𝜎) - 1$ .

Case IV: one endpoint of $𝜏$ lies in $𝜎_{𝑖}$ and the other is a fixed point of $𝜎$ . Similarly, this case also gives $𝑐 (𝜎 𝜏) = 𝑐 (𝜎) - 1$ .⁠
Proof. Note that $𝑐 (𝑒) = 𝑛$ , and $𝑐 (𝑥)$ changes by $\pm 1$ each time we multiply $𝑥$ by a transposition. So the minimum number of transpositions needed to express $𝜎$ as a product of transpositions is no less than $𝑛 - 𝑐 (𝜎)$ .

On the other hand, write
$𝜎 = \prod_{𝑖 = 1}^{𝑘} 𝜎_{𝑖} = \prod_{𝑖 = 1}^{𝑘} \prod_{𝑗 = 1}^{𝑟_{𝑖} - 1} (𝑎_{𝑖 𝑗} 𝑎_{𝑖, 𝑗 + 1}),$
there are exactly
$𝑟_{1} + 𝑟_{2} + \dots + 𝑟_{𝑘} - 𝑘 = 𝑛 - 𝑐 (𝜎)$
transpositions.⁠

2 Normal Subgroups, Quotient Groups, and Commutators

Problem 2.1

Let $𝐺$ be a group and let $𝐴 ⊴ 𝐺$ . Define

[𝐴, 𝐺] = ⟨ 𝑎^{- 1} 𝑔^{- 1} 𝑎 𝑔 | 𝑎 \in 𝐴, 𝑔 \in 𝐺 ⟩ .

Prove that

[𝐴, 𝐺] ⊴ 𝐺

Proof. For every $𝑎^{- 1} 𝑔^{- 1} 𝑎 𝑔 \in [𝐴, 𝐺]$ and $𝑔^{'} \in 𝐺$ , $𝑔^{' - 1} 𝑎 𝑔^{'} \in 𝐴$ because $𝐴 ⊴ 𝐺$ . Hence

\begin{aligned} 𝑔^{' - 1} 𝑎^{- 1} 𝑔^{- 1} 𝑎 𝑔 𝑔^{'} \\ = & (𝑔^{' - 1} 𝑎^{- 1} 𝑔^{'}) (𝑔^{' - 1} 𝑔^{- 1} 𝑔^{'}) (𝑔^{' - 1} 𝑎 𝑔^{'}) (𝑔^{' - 1} 𝑔 𝑔^{'}) \\ = & {(𝑔^{' - 1} 𝑎 𝑔^{'})}^{- 1} {(𝑔^{' - 1} 𝑔 𝑔^{'})}^{- 1} (𝑔^{' - 1} 𝑎 𝑔^{'}) (𝑔^{' - 1} 𝑔 𝑔^{'}) \\ = & [𝑔^{' - 1} 𝑎 𝑔^{'}, 𝑔^{' - 1} 𝑔 𝑔^{'}] \in [𝐴, 𝐺] . \end{aligned}

Problem 2.2

Let $𝐷_{𝑛} = ⟨ 𝑟, 𝑠 | 𝑟^{𝑛} = 𝑠^{2} = 𝑟 𝑠 𝑟 𝑠 = 𝑒 ⟩$ be the dihedral group ( $𝑛 \geq 3$ ). Prove the following statements:

If $𝑛$ is odd, then $[𝐷_{𝑛}, 𝐷_{𝑛}] = ⟨ 𝑟 ⟩ = ⟨ 𝑟^{2} ⟩$ ;
If $𝑛$ is even, then $[𝐷_{𝑛}, 𝐷_{𝑛}] = ⟨ 𝑟^{2} ⟩$ .

Proof. $𝐷_{𝑛} / ⟨ 𝑟 ⟩ = {𝑒, 𝑠}$ is abelian, so $[𝐷_{𝑛}, 𝐷_{𝑛}] \subseteq ⟨ 𝑟 ⟩$ . ⁹⁹ Recall that $[𝐺, 𝐺]$ is the $\subseteq$ -minimum of all $𝑁 ⊴ 𝐺$ such that $𝐻 / 𝑁$ is abelian. Furthermore, we have $⟨ 𝑟^{2} ⟩ \subseteq [𝐷_{𝑛}, 𝐷_{𝑛}] \subseteq ⟨ 𝑟 ⟩$ because $𝑟^{2} = {(𝑠 𝑟 𝑠)}^{- 1} 𝑟 = 𝑠^{- 1} 𝑟^{- 1} 𝑠 𝑟 = [𝑠, 𝑟] \in [𝐷_{𝑛}, 𝐷_{𝑛}]$ .

If $𝑛$ is odd, $⟨ 𝑟 ⟩ = ⟨ 𝑟^{2} ⟩$ , so $⟨ 𝑟^{2} ⟩ = [𝐷_{𝑛}, 𝐷_{𝑛}] = ⟨ 𝑟 ⟩$ ;
If $𝑛$ is even, $𝐷_{𝑛} / ⟨ 𝑟^{2} ⟩ = {𝑒, 𝑟, 𝑠, 𝑟 𝑠} ≅ 𝐶_{2} \times 𝐶_{2}$ is abelian, hence $[𝐷_{𝑛}, 𝐷_{𝑛}] \subseteq ⟨ 𝑟^{2} ⟩$ , then $[𝐷_{𝑛}, 𝐷_{𝑛}] = ⟨ 𝑟^{2} ⟩$ .⁠

Problem 2.3

Let

𝑆

be a set of representatives of left cosets of a subgroup

𝐻 \leq 𝐺

, i.e.

𝐺 = ⨆_{𝑎 \in 𝑆} 𝑎 𝐻

. Prove that

𝑆^{- 1}

is a set of representatives of right cosets of

𝐻

, i.e.

𝐺 = ⨆_{𝑎^{'} \in 𝑆^{- 1}} 𝐻 𝑎^{'}

Proof.

𝐺 = 𝐺^{- 1} = ⨆_{𝑎 \in 𝑆} {(𝑎 𝐻)}^{- 1} = ⨆_{𝑎 \in 𝑆} 𝐻^{- 1} 𝑎^{- 1} = ⨆_{𝑎 \in 𝑆} 𝐻 𝑎^{- 1} = ⨆_{𝑎^{'} \in 𝑆^{- 1}} 𝐻 𝑎^{'} .

Problem 2.4

Let

𝐺

be a group and let

𝑁 ⊴ 𝐺

. If

𝐺 / 𝑁

is finite and

| 𝐺 / 𝑁 | = 𝑛

, prove that

𝑎^{𝑛} \in 𝑁

for every

𝑎 \in 𝐺

Proof. Note that $𝑎 𝑁 \in 𝐺 / 𝑁$ and $𝑜 (𝑎 𝑁) | | 𝐺 / 𝑁 | = 𝑛$ , so ${(𝑎 𝑁)}^{𝑛} = 𝑎^{𝑛} 𝑁 = 𝑁$ , which implies $𝑎^{𝑛} \in 𝑁$ .⁠

Problem 2.5

Let

𝐺

be a finite group and let

𝐾 ⊴ 𝐻 ⊴ 𝐺

. If

gcd {| 𝐾 |, | 𝐻 / 𝐾 |} = 1

, prove that

𝐾 ⊴ 𝐺

Proof. For every $𝑔 \in 𝐺$ , we shall prove $𝐾 = 𝑔^{- 1} 𝐾 𝑔$ . Clearly, $| 𝐾 | = | 𝑔^{- 1} 𝐾 𝑔 |$ . ¹⁰¹⁰ One can consider the bijection $𝜑 : 𝑘 \mapsto 𝑔^{- 1} 𝑘 𝑔$ . It suffices to prove $𝑔^{- 1} 𝐾 𝑔 \subseteq 𝐾$ .

For every $𝑔^{- 1} 𝑘 𝑔 \in 𝑔^{- 1} 𝐾 𝑔$ , there exists $ℎ \in 𝐻$ such that $𝑘 𝑔 = 𝑔 ℎ$ since $𝑘 \in 𝐻 ⊴ 𝐺$ . So $𝑔^{- 1} 𝑘 𝑔 = 𝑔^{- 1} 𝑔 ℎ = ℎ \in 𝐻$ , which implies $𝑔^{- 1} 𝐾 𝑔 \leq 𝐻$ .

Furthermore, for every $ℎ \in 𝐻$ , there exists $ℎ^{'} \in 𝐻$ such that $𝑔 ℎ = ℎ^{'} 𝑔$ , so $ℎ^{- 1} 𝑔^{- 1} 𝐾 𝑔 ℎ = {(𝑔 ℎ)}^{- 1} 𝐾 (𝑔 ℎ) = {(ℎ^{'} 𝑔)}^{- 1} 𝐾 ℎ^{'} 𝑔 = 𝑔^{- 1} ℎ^{' - 1} 𝐾 ℎ^{'} 𝑔 = 𝑔^{- 1} 𝐾 𝑔$ , which implies $𝑔^{- 1} 𝐾 𝑔 ⊴ 𝐻$ .

Therefore, $𝐾 (𝑔^{- 1} 𝐾 𝑔)$ is normal in $𝐻$ , and $𝐾$ is normal in $𝐾 (𝑔^{- 1} 𝐾 𝑔)$ . ¹¹¹¹ Recall that if $𝐻 ⊴ 𝐺$ and $𝐾 \leq 𝐺$ , then $𝐻 ⊴ 𝐻 𝐾 \leq 𝐺$ ; When $𝐾 ⊴ 𝐺$ , $𝐻 𝐾$ is also normal in $𝐺$ . So we have

| 𝐾 (𝑔^{- 1} 𝐾 𝑔) | | | 𝐻 | \Rightarrow \frac{| 𝐾 (𝑔^{- 1} 𝐾 𝑔) |}{| 𝐾 |} | \frac{| 𝐻 |}{| 𝐾 |} .

On the other hand,

| 𝐾 (𝑔^{- 1} 𝐾 𝑔) | = \frac{| 𝐾 | \cdot | 𝑔^{- 1} 𝐾 𝑔 |}{| 𝐾 \cap 𝑔^{- 1} 𝐾 𝑔 |} = \frac{{| 𝐾 |}^{2}}{| 𝐾 \cap 𝑔^{- 1} 𝐾 𝑔 |},

thus

\frac{| 𝐾 (𝑔^{- 1} 𝐾 𝑔) |}{| 𝐾 |} = \frac{| 𝐾 |}{| 𝐾 \cap 𝑔^{- 1} 𝐾 𝑔 |} \Rightarrow \frac{| 𝐾 (𝑔^{- 1} 𝐾 𝑔) |}{| 𝐾 |} | | 𝐾 | .

So $\frac{| 𝐾 (𝑔^{- 1} 𝐾 𝑔) |}{| 𝐾 |}$ is a common divisor of $| 𝐾 |$ and $\frac{| 𝐻 |}{| 𝐾 |}$ . Given

gcd {| 𝐾 |, \frac{| 𝐻 |}{| 𝐾 |}} = gcd {| 𝐾 |, | 𝐻 / 𝐾 |} = 1,

then $| 𝐾 (𝑔^{- 1} 𝐾 𝑔) | = | 𝐾 |$ , hence $𝐾 = 𝐾 (𝑔^{- 1} 𝐾 𝑔)$ because $𝐾 \subseteq 𝐾 (𝑔^{- 1} 𝐾 𝑔)$ , which implies $𝑔^{- 1} 𝐾 𝑔 \subseteq 𝐾$ .⁠

Caution

One may want to do something with the normality of $𝐾 \cap (𝑔^{- 1} 𝐾 𝑔)$ , but the division

\frac{| 𝐾 \cap (𝑔^{- 1} 𝐾 𝑔) |}{| 𝐾 |} | \frac{| 𝐻 |}{| 𝐾 |}

is invalid.

Problem 2.6

Prove the following statements.

For every positive integer $𝑛$ , $⟨ \frac{1}{𝑛} + ℤ ⟩$ is the unique subgroup of $ℚ / ℤ$ of order $𝑛$ .
Every subgroup of $ℚ / ℤ$ generated by finitely many elements is cyclic.

Proof. Let $𝐺$ be a subgroup of $ℚ / ℤ$ of order $𝑛$ . For each $𝑞 \in 𝐺$ , if $𝑞 = 0$ , then $𝑞 \in ⟨ \frac{1}{𝑛} + ℤ ⟩$ . Otherwise, write $𝑞 = \frac{𝑎}{𝑏} \neq 0$ with $𝑎 \in ℤ^{*}, 𝑏 \in ℕ^{+}$ . Since $𝑜 (𝑞) | 𝑛$ ,
$𝑛 𝑞 = (\frac{𝑛 𝑎}{𝑏}) = 0 \Rightarrow \frac{𝑛 𝑎}{𝑏} \in ℤ .$
Thus
$𝑞 = (\frac{1}{𝑛}) \cdot (\frac{𝑛 𝑎}{𝑏}) = (\frac{𝑛 𝑎}{𝑏}) (\frac{1}{𝑛}) \in ⟨ \frac{1}{𝑛} + ℤ ⟩,$
Hence $𝐺 \leq ⟨ \frac{1}{𝑛} + ℤ ⟩$ and furthermore $𝐺 = ⟨ \frac{1}{𝑛} + ℤ ⟩$ because $| 𝐺 | = | ⟨ \frac{1}{𝑛} + ℤ ⟩ | = 𝑛$ .⁠
Proof. Let $𝐺$ = $⟨ 𝑞_{1}, \dots, 𝑞_{𝑛} ⟩$ , $𝑞_{𝑖} \in ℚ$ . Write $𝑞_{𝑖} = \frac{𝑎_{𝑖}}{𝑏_{𝑖}}$ , $𝑎_{𝑖} \in ℤ, 𝑏_{𝑖} \in ℕ^{+}$ .

Let $𝑚 = lcm {𝑏_{1}, \dots, 𝑏_{𝑛}} \in ℕ^{+}$ , then
$𝑞_{𝑖} = (\frac{𝑎_{𝑖} 𝑚}{𝑏_{𝑖} 𝑚}) = (\frac{𝑎_{𝑖} 𝑚}{𝑏_{𝑖}}) (\frac{1}{𝑚}) \in ⟨ \frac{1}{𝑚} + ℤ ⟩,$
so $𝐺 \leq ⟨ \frac{1}{𝑚} + ℤ ⟩$ is cyclic. ¹²¹² Recall that all subgroups of a cyclic group is cyclic.⁠

Problem 2.7

Let $𝐺$ be a finite group and let $𝐻$ be a proper subgroup of $𝐺$ . Subgroups of the form $𝑔 𝐻 𝑔^{- 1} (𝑔 \in 𝐺)$ are called conjugates of $𝐻$ . Prove the following statements.

$𝐻$ has $[𝐺 : 𝑁_{𝐺} (𝐻)]$ conjugates.
The set of all conjugates of $𝐻$ cannot cover the whole group $𝐺$ .

Proof. Let $𝑥, 𝑦 \in 𝐺$ . Note that
$\begin{aligned} 𝑥 𝐻 𝑥^{- 1} = 𝑦 𝐻 𝑦^{- 1} & \Leftrightarrow 𝐻 = {(𝑦^{- 1} 𝑥)}^{- 1} 𝐻 (𝑦^{- 1} 𝑥) \\ \Leftrightarrow 𝑦^{- 1} 𝑥 \in 𝑁_{𝐺} (𝐻) \\ \Leftrightarrow 𝑥 𝑁_{𝐺} (𝐻) = 𝑦 𝑁_{𝐺} (𝐻) . \end{aligned}$
¹³¹³ Recall that two cosets $𝑥 𝐻$ and $𝑦 𝐻$ are either disjoint or equal, and $𝑥 𝐻 = 𝑦 𝐻 \Leftrightarrow 𝑥 𝑦^{- 1} \in 𝐻$ . By the definition of group index, $𝐻$ has $[𝐺 : 𝑁_{𝐺} (𝐻)]$ distinct conjugates.⁠
Proof. If $𝑁_{𝐺} (𝐻) = 𝐺$ , $𝐻$ itself is the only conjugate of $𝐻$ , and it does not cover $𝐺$ because $𝐻$ is a proper subgroup.

Otherwise, $𝑁_{𝐺} (𝐻)$ is a proper subgroup of $𝐺$ , so $| 𝑁_{𝐺} (𝐻) | < | 𝐺 |$ . Let $𝑆$ denote the set of representatives for the left cosets of $𝑁_{𝐺} (𝐻)$ in $𝐺$ , then
$𝐾 = {𝑔 𝐻 𝑔^{- 1} | 𝑔 \in 𝑆}$
is the set of conjugates of $𝐻$ . We shall prove $| ⋃ 𝐾 | < | 𝐺 |$ .

Since $| 𝑔 𝐻 𝑔^{- 1} | = | 𝐻 |$ and each conjugate contains the identity:
$\begin{aligned} | ⋃ 𝐾 | & = | ⋃ {𝑒, ⋃_{𝑔 \in 𝑆} (𝑔 𝐻 𝑔^{- 1} \ {𝑒})} | \\ \leq 1 + \sum_{𝑔 \in 𝑆} (| 𝑔 𝐻 𝑔^{- 1} | - 1) \\ = 1 + | 𝑆 | \cdot (| 𝐻 | - 1) \\ = 1 + \frac{| 𝐺 | \cdot (| 𝐻 | - 1)}{| 𝑁_{𝐺} (𝐻) |} \\ < \frac{| 𝐺 |}{| 𝑁_{𝐺} (𝐻) |} + \frac{| 𝐺 | \cdot (| 𝐻 | - 1)}{| 𝑁_{𝐺} (𝐻) |} \\ = \frac{| 𝐺 | \cdot | 𝐻 |}{| 𝑁_{𝐺} (𝐻) |} \\ \leq | 𝐺 | . \end{aligned}$

Problem 2.8

Let $𝐺$ be a finite group and let $𝐻, 𝐾$ be subgroups of $𝐺$ . If

gcd {[𝐺 : 𝐻], [𝐺 : 𝐾]} = 1,

prove that

𝐺 = 𝐻 𝐾

Proof. Recall that

| 𝐻 \cap 𝐾 | = \frac{| 𝐻 | \cdot | 𝐾 |}{| 𝐻 𝐾 |},

then

\frac{| 𝐺 |}{| 𝐻 \cap 𝐾 |} = \frac{{| 𝐺 |}^{2}}{| 𝐻 | \cdot | 𝐾 |} \cdot \frac{| 𝐻 𝐾 |}{| 𝐺 |} .

Since $𝐻 \cap 𝐾 \leq 𝐻 \leq 𝐺$ ,

| 𝐻 \cap 𝐾 | | | 𝐻 |, | 𝐻 | | | 𝐺 |,

\frac{| 𝐺 |}{| 𝐻 |} | \frac{| 𝐺 |}{| 𝐻 \cap 𝐾 |} .

Similarly,

\frac{| 𝐺 |}{| 𝐾 |} | \frac{| 𝐺 |}{| 𝐻 \cap 𝐾 |} .

Given

gcd {[𝐺 : 𝐻], [𝐺 : 𝐾]} = gcd {\frac{| 𝐺 |}{| 𝐻 |}, \frac{| 𝐺 |}{| 𝐾 |}} = 1,

we have

\frac{{| 𝐺 |}^{2}}{| 𝐻 | \cdot | 𝐾 |} | \frac{| 𝐺 |}{| 𝐻 \cap 𝐾 |} .

Hence

\frac{| 𝐺 |}{| 𝐻 \cap 𝐾 |} / \frac{{| 𝐺 |}^{2}}{| 𝐻 | \cdot | 𝐾 |} = \frac{| 𝐻 𝐾 |}{| 𝐺 |} \in ℤ \Rightarrow | 𝐺 | | | 𝐻 𝐾 |,

which implies $𝐺 = 𝐻 𝐾$ because $𝐻 𝐾 \subseteq 𝐺$ .⁠

Caution

Recall that

𝐻 𝐾

is not necessarily a subgroup, do not claim something like

𝐻 \leq 𝐻 𝐾 \leq 𝐺

Problem 2.9

Let

𝐺

be a subgroup of

𝑆_{2024}

, and

𝐺

contains an

𝑛

-cycle, where

𝑛 \geq 4

is even. Prove that

𝐺

is not simple.

Proof. Consider the intersection $𝐻 = 𝐺 \cap 𝐴_{2024} ⊴ 𝐺$ . ¹⁴¹⁴ Normality is ensured by the isomorphic theorems.

Let $𝜎 \in 𝐺$ be an $𝑛$ -cycle with $𝑛 \geq 4$ even, then $𝐻$ is proper because $𝜎$ is odd, so $𝜎 \notin 𝐻$ . $𝐻$ is also nontrivial because $𝜎^{2}$ is even. ¹⁵¹⁵ $𝜎^{2} \neq 𝑒$ is a product of two $\frac{𝑛}{2}$ cycles, so it can be expressed as a product of $2 \cdot (\frac{𝑛}{2} - 1)$ transpositions.⁠

Problem 2.10

Find the structure of the following two subgroups of $𝑆_{5}$ :

$⟨ (12) (34), (12345) ⟩$ ;
$⟨ (12) (35), (12345) ⟩$ .

Solution. Let $𝐻 = ⟨ (12) (34), (12345) ⟩$ . Note that $𝐻 \leq 𝐴_{5}$ , ¹⁶¹⁶ The generators $(12) (34)$ and $(12345)$ are both even. so $| 𝐻 | | | 𝐴_{5} | = 60$ .

Since
$(12) (34) \in 𝐻, (12345) \in 𝐻, (12) (34) (12345) = (245) \in 𝐻,$
we have
$lcm {𝑜 ((12) (34)), 𝑜 ((12345)), 𝑜 ((245))} = 30 | | 𝐻 | .$
Thus $| 𝐻 | = 30$ or $| 𝐻 | = 60$ . Index $2$ would force $𝐻 ⊴ 𝐴_{5}$ , impossible. Hence $𝐻 = 𝐴_{5}$ .
Solution. Let
$𝑠 = (12) (35), 𝑟 = (12345), 𝐺 = ⟨ 𝑟, 𝑠 ⟩ .$
Note that
$𝑠^{2} = 𝑟^{5} = 𝑟 𝑠 𝑟 𝑠 = 𝑒,$
so $𝐺 ≅ 𝐷_{5}$ .

Problem 2.11

Let $𝐴_{\infty} = ⋃_{𝑛 \geq 2} 𝐴_{𝑛}$ . Prove the following statements.

$𝐴_{\infty}$ has a natural group structure such that $𝐴_{𝑛} < 𝐴_{\infty}$ for every $𝑛 \geq 2$ .
$𝐴_{\infty}$ is simple.

Proof. For every $𝑥 \in 𝐴_{\infty}$ , let $𝑁_{𝑥}$ denote the nonempty set ${𝑛 \in ℕ^{+} | 𝑥 \in 𝐴_{𝑛}}$ because $𝑥$ belongs to at least one $𝐴_{𝑛}$ , and let $𝑛 (𝑥) = min 𝑁_{𝑥}$ . Define the inverse of $𝑥$ to be $𝑥^{- 1}$ in $𝐴_{𝑛 (𝑥)}$ .

For every $𝑥, 𝑦 \in 𝐴_{\infty}$ , define the product $𝑥 \cdot 𝑦$ to be $𝑥 𝑦$ in $𝐴_{max {𝑛 (𝑥), 𝑛 (𝑦)}}$ .

Clearly, $𝐴_{\infty}$ forms a group under these operations with identity $(1)$ . And there is an infinite chain of proper subgroups:
$𝐴_{2} < 𝐴_{3} < \dots < 𝐴_{𝑛} < \dots < 𝐴_{\infty} .$
Proof. Let $𝑁$ be a normal subgroup of $𝐴_{\infty}$ . We shall prove $𝑁 = {𝑒}$ or $𝑁 = 𝐴_{\infty}$ .

For each $𝑛 \geq 5$ , consider the intersection $𝑁 \cap 𝐴_{𝑛} ⊴ 𝐴_{𝑛}$ . ¹⁷¹⁷ $𝑁 ⊴ 𝐴_{\infty}$ and $𝐴_{𝑛} \leq 𝐴_{\infty}$ . Because of the simplicity of $𝐴_{𝑛}$ , either $𝑁 \cap 𝐴_{𝑛} = {𝑒}$ or $𝑁 \cap 𝐴_{𝑛} = 𝐴_{𝑛}$ .

If $𝑁 \cap 𝐴_{5} = {𝑒}$ , we actually have
$𝑁 \cap 𝐴_{𝑛} = {𝑒} for each 𝑛 \geq 5,$
¹⁸¹⁸ If there exists $𝑛 > 5$ such that $𝑁 \cap 𝐴_{𝑛} = 𝐴_{𝑛}$ , then $𝐴_{5} \subseteq 𝐴_{𝑛} \subseteq 𝑁$ , contradicting to $𝑁 \cap 𝐴_{5} = {𝑒}$ . so
$𝑁 = 𝑁 \cap 𝐴_{\infty} = 𝑁 \cap (⋃_{𝑛 = 5}^{\infty} 𝐴_{𝑛}) = ⋃_{𝑛 = 5}^{\infty} (𝑁 \cap 𝐴_{𝑛}) = {𝑒} .$
Similarly, if $𝑁 \cap 𝐴_{5} = 𝐴_{5}$ , we have
$𝑁 \cap 𝐴_{𝑛} = 𝐴_{𝑛} for each 𝑛 \geq 5,$
so
$𝑁 = 𝑁 \cap 𝐴_{\infty} = 𝑁 \cap (⋃_{𝑛 = 5}^{\infty} 𝐴_{𝑛}) = ⋃_{𝑛 = 5}^{\infty} (𝑁 \cap 𝐴_{𝑛}) = ⋃_{𝑛 = 5}^{\infty} 𝐴_{𝑛} = 𝐴_{\infty} .$

Problem 2.12

Let $𝑛 \geq 2$ be a positive integer.

Let $𝐺_{𝑛} = [𝑆_{𝑛}, 𝑆_{𝑛}]$ . Prove that $𝐺_{𝑛} = 𝐴_{𝑛}$ for $𝑛 \geq 3$ .
Find $[𝐴_{𝑛}, 𝐴_{𝑛}]$ .

Proof. Since $𝑆_{𝑛} / 𝐴_{𝑛} ≅ 𝐶_{2}$ is abelian, $𝐺_{𝑛} = [𝑆_{𝑛}, 𝑆_{𝑛}] \subseteq 𝐴_{𝑛}$ . For each $3 \leq 𝑖 \leq 𝑛$ ,
$[(12), (1 𝑖)] = {(12)}^{- 1} {(1 𝑖)}^{- 1} (12) (1 𝑖) = (12 𝑖) \in 𝐺_{𝑛},$
so $𝐺_{𝑛}$ contains all the generators of $𝐴_{𝑛}$ ¹⁹¹⁹ Recall that $𝐴_{𝑛} = ⟨ (123), (124), \dots, (12 𝑛) ⟩$ . , then $𝐴_{𝑛} \subseteq 𝐺_{𝑛}$ . Hence $𝐺_{𝑛} = 𝐴_{𝑛}$ .⁠
Solution.
1. If $𝑛 \leq 3$ , $𝐴_{𝑛}$ is abelian, then $[𝐴_{𝑛}, 𝐴_{𝑛}] = {𝑒}$ .
2. If $𝑛 \geq 5$ , $𝐴_{𝑛}$ is simple, so $[𝐴_{𝑛}, 𝐴_{𝑛}] = {𝑒}$ or $[𝐴_{𝑛}, 𝐴_{𝑛}] = 𝐴_{𝑛}$ . Compute
  $[(123), (124)] = {(123)}^{- 1} {(124)}^{- 1} (123) (124) = (12) (34) \in [𝐴_{𝑛}, 𝐴_{𝑛}],$ so $[𝐴_{𝑛}, 𝐴_{𝑛}] = 𝐴_{𝑛}$ .
3. If $𝑛 = 4$ , $𝑉_{4}$ is normal in $𝐴_{4}$ and $𝐴_{4} / 𝑉_{4} ≅ 𝐶_{3}$ is abelian, so $[𝐴_{4}, 𝐴_{4}] \subseteq 𝑉_{4}$ . ²⁰²⁰ Recall that $𝑉_{4} = {𝑒, (12) (34), (13) (24), (14) (23)}$ which is isomorphic to $⟨ 𝑎, 𝑏 | 𝑎^{2} = 𝑏^{2} = {(𝑎 𝑏)}^{2} = 𝑒 ⟩$ .
  
  Also compute $[(123), (134)] = (14) (23) \in [𝐴_{4}, 𝐴_{4}]$ , so $𝑉_{4} = ⟨ (12) (34), (14) (23) ⟩ \subseteq [𝐴_{4}, 𝐴_{4}]$ .
  
  Therefore, $[𝐴_{4}, 𝐴_{4}] = 𝑉_{4}$ .

3 Normal Subgroups and Automorphisms

Problem 3.1

Let $𝑁$ be a normal subgroup of $𝐴_{4}$ that contains a $3$ -cycle. Prove that $𝑁 = 𝐴_{4}$ .
Find all normal subgroups of $𝐴_{4}$ .

Proof. Since $3 | | 𝑁 |$ and $| 𝑁 | | | 𝐴_{4} | = 12$ , $| 𝑁 | \in {3, 6, 12}$ .
1. If $| 𝑁 | = 12$ , we are done.
2. If $| 𝑁 | = 3$ , $𝑁 = ⟨ 𝜎 ⟩$ where $𝜎 = (𝑎 𝑏 𝑐)$ . Let ${𝑑} = {1, 2, 3, 4} \ {𝑎, 𝑏, 𝑐}$ and let $𝜏 = (𝑎 𝑑 𝑏) \in 𝐴_{4}$ . Compute
  $𝜏 𝜎 𝜏^{- 1} = (𝜏 (𝑎) 𝜏 (𝑏) 𝜏 (𝑐)) = (𝑑 𝑎 𝑐) \notin 𝑁,$ contradicting the normality of $𝑁$ .
3. If $| 𝑁 | = 6$ , $𝐴_{4} / 𝑁 ≅ 𝐶_{2}$ is abelian, so $𝑉_{4} = [𝐴_{4}, 𝐴_{4}] \leq 𝑁$ . But $| 𝑉_{4} | = 4 ∤ 6$ , contradiction.
⁠
Solution. Let $𝑁 ⊴ 𝐴_{4}$ , then $| 𝑁 | \in {1, 2, 3, 6, 12}$ . $| 𝑁 | = 1$ and $| 𝑁 | = 12$ are trivial cases, and we have proved $| 𝑁 | = 3$ and $| 𝑁 | = 6$ are impossible.
1. If $| 𝑁 | = 2$ , then $𝑁 = ⟨ (𝑎 𝑏) (𝑐 𝑑) ⟩$ , but
  $(𝑎 𝑏 𝑐) (𝑎 𝑏) (𝑐 𝑑) {(𝑎 𝑏 𝑐)}^{- 1} = (𝑎 𝑑) (𝑏 𝑐) \notin 𝑁,$ contradiction.
2. If $| 𝑁 | = 4$ , $𝐴_{4} / 𝑁 ≅ 𝐶_{3}$ is abelian, so $𝑉_{4} = [𝐴_{4}, 𝐴_{4}] \leq 𝑁$ , which implies $𝑁 = 𝑉_{4}$ .
In conclusion, normal subgroups of $𝐴_{4}$ are ${𝑒}, 𝑉_{4}, 𝐴_{4}$ .

Problem 3.2

Find all normal subgroups of

𝑆_{𝑛} (𝑛 \geq 1)

Lemma 3.3

𝐺

is a group with trivial center, then it cannot have a normal subgroup of order

2

Proof. Let $𝑁 ⊴ 𝐺$ and $| 𝑁 | = 2$ , then $𝑁 = {𝑒, 𝑥}$ with $𝑥^{2} = 𝑒$ .

For every $𝑔 \in 𝐺$ ,

𝑔 𝑁 𝑔^{- 1} = {𝑒, 𝑔 𝑥 𝑔^{- 1}} = 𝑁 = {𝑒, 𝑥} \Rightarrow 𝑔 𝑥 𝑔^{- 1} = 𝑥

by the normality of $𝑁$ .

This implies

𝑥 \in 𝑍 (𝐺) \Rightarrow 𝑥 = 𝑒 \Rightarrow 𝑁 = {𝑒},

contradiction.⁠

Solution of Problem 3.2.

$𝑛 = 1$ : ${𝑒}$ .
$𝑛 = 2$ : ${𝑒}, 𝑆_{2}$ .
$𝑛 = 3$ : ${𝑒}, 𝐴_{3}, 𝑆_{3}$ .
$𝑛 = 4$ : ${𝑒}, 𝑉_{4}, 𝐴_{4}, 𝑆_{4}$ .
$𝑛 \geq 5$ : Let $𝑁 ⊴ 𝑆_{𝑛}$ . Consider the intersection $𝐻 = 𝑁 \cap 𝐴_{𝑛} ⊴ 𝐴_{𝑛}$ , then $𝐻 = {𝑒}$ or $𝐻 = 𝐴_{𝑛}$ since $𝐴_{𝑛}$ is simple.
1. $𝐻 = {𝑒}$ implies every nontrivial permutation in $𝑁$ is odd. Let $𝜑 : 𝑁 \to 𝑆_{𝑛} / 𝐴_{𝑛}$ defined as $𝑔 \mapsto 𝑔 𝐴_{𝑛}$ , $Ker 𝜑 = 𝑁 \cap 𝐴_{𝑛} = {𝑒}$ , so
  $| 𝑁 | = | Im (𝜑) | \leq | 𝑆_{𝑛} / 𝐴_{𝑛} | = 2 .$ Recall that $𝑆_{𝑛}$ has trivial center when $𝑛 \geq 3$ . By Lemma 3.3, $𝑁 = {𝑒}$ .
2. $𝐻 = 𝐴_{𝑛}$ implies $𝐴_{𝑛} \leq 𝑁$ . But $𝐴_{𝑛}$ is already the maximal proper subgroup of $𝑆_{𝑛}$ , so $𝑁 = 𝐴_{𝑛}$ or $𝑁 = 𝑆_{𝑛}$ .
Hence there are three normal subgroups: ${𝑒}, 𝐴_{𝑛}, 𝑆_{𝑛}$ .

Problem 3.4

For

𝑛 \geq 2

, determine all integers

𝑖

with

2 \leq 𝑖 \leq 𝑛

such that

𝑆_{𝑛} = ⟨ (1 𝑖), (12 \dots 𝑛) ⟩

Solution. The indices in our solution should wrap around $1$ and $𝑛$ .

Note that

{(12 \dots 𝑛)}^{𝑘} (1 𝑖) {(12 \dots 𝑛)}^{- 𝑘} = (1 + 𝑘, 1 + 𝑘 + 𝑖 - 1),

so we can get transpositions of the form $(𝑎, 𝑎 + 𝑖 - 1)$ by conjugations of the two generators.

Let $𝐺 = (𝑉, 𝐸)$ be an undirected graph with $𝑉 = {1, 2, \dots, 𝑛}$ and $𝐸 = {(𝑎, 𝑎 + 𝑖 - 1) | 1 \leq 𝑎 \leq 𝑛}$ . The connected components of $𝐺$ are cosets of $⟨ 𝑖 - 1 ⟩$ in $ℤ / 𝑛 ℤ$ , hence

\begin{aligned} (1 𝑖) and (12 \dots 𝑛) generates 𝑆_{𝑛} & \Leftrightarrow 𝐺 is connected \\ \Leftrightarrow ⟨ 𝑖 - 1 ⟩ = ℤ / 𝑛 ℤ \\ \Leftrightarrow gcd {𝑛, 𝑖 - 1} = 1 . \end{aligned}

Problem 3.5

Let

𝑛

be a positive integer. Prove that

𝜎, 𝜏 \in 𝑆_{𝑛}

are conjugate if and only if they have the same type.

Proof. Recall that

𝑔 (𝑎_{1} 𝑎_{2} \dots 𝑎_{𝑘}) 𝑔^{- 1} = (𝑔 (𝑎_{1}) 𝑔 (𝑎_{2}) \dots 𝑔 (𝑎_{𝑘})), \forall 𝑔 \in 𝑆_{𝑛},

so conjugation of permutations preserves the cycle type. It remains to prove the converse.

Let $𝜎, 𝜏 \in 𝑆_{𝑛}$ with the same type. Decompose them into disjoint cycles (including trivial ones):

𝜎 = \prod_{𝑖 = 1}^{𝑘} 𝜎_{𝑖} = \prod_{𝑖 = 1}^{𝑘} (𝑎_{𝑖 1} 𝑎_{𝑖 2} \dots 𝑎_{𝑖 𝑟_{𝑖}}), 𝜏 = \prod_{𝑖 = 1}^{𝑘} 𝜏_{𝑖} = \prod_{𝑖 = 1}^{𝑘} (𝑏_{𝑖 1} 𝑏_{𝑖 2} \dots 𝑏_{𝑖 𝑟_{𝑖}}) .

Define $𝜋 : 𝑎_{𝑖 𝑗} \mapsto 𝑏_{𝑖 𝑗}$ , then $𝜋 \in 𝑆_{𝑛}$ and $𝜋 𝜎 𝜋^{- 1} = 𝜏$ .⁠

Problem 3.6

Let

𝑓 : 𝐺 \to 𝐻

be a group homomorphism and let

𝑎 \in 𝐺

be an element of finite order. Prove that the order of

𝑓 (𝑎)

divides the order of

𝑎

Proof.

{(𝑓 (𝑎))}^{𝑜 (𝑎)} = 𝑓 (𝑎^{𝑜 (𝑎)}) = 𝑓 (𝑒) = 𝑒 \Rightarrow 𝑜 (𝑓 (𝑎)) | 𝑜 (𝑎) .

Problem 3.7

Let $𝐺$ be a group and define $𝑓 : 𝐺 \to 𝐺$ to be the map $𝑥 \mapsto 𝑥^{- 1}$ .

Give a necessary and sufficient condition for $𝑓$ to be a group homomorphism.
If $𝐺$ is finite and $| Aut (𝐺) | = 1$ , prove that $| 𝐺 | \leq 2$ .

Solution.
$\begin{aligned} 𝑓 is a group homomorphism & \Leftrightarrow \forall 𝑎, 𝑏 \in 𝐺, 𝑓 (𝑎) 𝑓 (𝑏) = 𝑓 (𝑎 𝑏) \\ \Leftrightarrow \forall 𝑎, 𝑏 \in 𝐺, 𝑎^{- 1} 𝑏^{- 1} = {(𝑎 𝑏)}^{- 1} = 𝑏^{- 1} 𝑎^{- 1} \\ \Leftrightarrow 𝐺 is abelian, equivalently 𝑍 (𝐺) = 𝐺 . \end{aligned}$
Proof. Since $Inn (𝐺) ⊴ Aut (𝐺)$ and $| Aut (𝐺) | = 1$ , $| Inn (𝐺) | = 1$ .

Consider the homomorphism $𝜑 : 𝑔 \mapsto 𝜃_{𝑔} = (𝑥 \mapsto 𝑔 𝑥 𝑔^{- 1})$ , then ²¹²¹ Recall that $Ker 𝜑 = 𝑍 (𝐺)$ and $Im 𝜑 = Inn (𝐺)$ .
$𝐺 / 𝑍 (𝐺) ≅ Inn (𝐺) \Rightarrow \frac{| 𝐺 |}{| 𝑍 (𝐺) |} = | Inn (𝐺) | = 1 \Rightarrow 𝐺 = 𝑍 (𝐺),$
which makes $𝑓$ a homomorphism. Furthermore, $𝑓 \in Aut (𝐺)$ , so $𝑓 = id$ . We have
$𝑥 = 𝑥^{- 1} for each 𝑥 \in 𝐺$
and every element has order at most $2$ .

If $| 𝐺 | > 2$ , then $𝐺 ≅ {(ℤ / 2 ℤ)}^{𝑑}, 𝑑 \geq 2$ . Clearly, such a group has nontrivial automorphisms, ²²²² e.g. swap two basis vectors. contradiction.⁠

Problem 3.8

Let

𝐺

be a group with trivial center, prove that

Aut (𝐺)

also has trivial center.

Proof. For every $𝜎 \in 𝑍 (Aut (𝐺))$ and $𝑥 \in 𝐺$ , let $𝜃_{𝑥} \in Aut (𝐺)$ denote the conjugation induced by $𝑥$ , i.e. $𝜃_{𝑥} : 𝑔 \mapsto 𝑥 𝑔 𝑥^{- 1}$ .

Note that $𝜎 𝜃_{𝑥} = 𝜃_{𝜎 (𝑥)} 𝜎$ , we have

𝜎 𝜃_{𝑥} = 𝜃_{𝜎 (𝑥)} 𝜎 = 𝜃_{𝑥} 𝜎 \Rightarrow 𝜃_{𝑥} = 𝜃_{𝜎 (𝑥)}

since $𝜎$ commutes with $𝜃_{𝑥}$ .

By injectivity of $𝑔 \mapsto 𝜃_{𝑔}$ ²³²³ The kernel $𝑍 (𝐺)$ is trivial.,

𝜃_{𝑥} = 𝜃_{𝜎 (𝑥)} \Rightarrow 𝜎 (𝑥) = 𝑥 \Rightarrow 𝜎 = id .

Hence $Aut (𝐺)$ has trivial center.⁠

Problem 3.9

Let

𝐺

be a finite group such that

| 𝐺 | = 𝑝_{1} 𝑝_{2} \dots 𝑝_{𝑠}

, where the

𝑝_{𝑖}

are distinct primes. If

𝐺

has

𝑛

normal subgroups, prove that

𝑛 \leq 2^{𝑠}

Note that $| 𝐺 |$ has exactly $2^{𝑠}$ positive divisors, so there would be two distinct normal subgroups sharing the same order if $𝑛 > 2^{𝑠}$ . If we can prove “identical orders imply identical groups”, then we are done.

Lemma 3.10

Let

𝐺

be a finite group such that

| 𝐺 | = 𝑝_{1} 𝑝_{2} \dots 𝑝_{𝑠}

, where the

𝑝_{𝑖}

are distinct primes. If

𝑁_{1}

and

𝑁_{2}

are normal subgroups of

𝐺

with the same order

𝑚

, then

𝑁_{1} = 𝑁_{2}

Proof. By induction on $𝑠$ . When $𝑠 = 1$ , $𝐺$ is cyclic with prime order, so either $𝑁_{1} = 𝑁_{2} = {𝑒}$ or $𝑁_{1} = 𝑁_{2} = 𝐺$ .

Next, we assume the lemma holds for $1, 2, \dots, 𝑠 - 1$ .

If $𝑚 = 1$ , $𝑁_{1} = 𝑁_{2} = {𝑒}$ .
If $𝑚 \geq 2$ and $𝑁_{1} 𝑁_{2} = 𝐺$ , we have
$| 𝐺 | = | 𝑁_{1} 𝑁_{2} | = \frac{| 𝑁_{1} | | 𝑁_{2} |}{| 𝑁_{1} \cap 𝑁_{2} |} = \frac{𝑚^{2}}{| 𝑁_{1} \cap 𝑁_{2} |}$ which forces $| 𝑁_{1} \cap 𝑁_{2} | = 𝑚$ because $| 𝐺 |$ is squarefree. Thus $𝑁_{1} \cap 𝑁_{2} = 𝑁_{1} = 𝑁_{2}$ .
If $𝑚 \geq 2$ and $𝑁_{1} 𝑁_{2} < 𝐺$ , since $𝑁_{1}$ and $𝑁_{2}$ normal in $𝑁_{1} 𝑁_{2}$ with the same order, and $| 𝑁_{1} 𝑁_{2} |$ is a product of $𝑘$ distinct primes ( $1 \leq 𝑘 \leq 𝑠 - 1$ ), we have $𝑁_{1} = 𝑁_{2}$ by the induction hypothesis.

⁠

Problem 3.11

For

𝑛 \geq 3

, find

| Aut (𝐷_{𝑛}) |

Problem 3.12

Let $𝜑 : 𝑆_{𝑛} \to 𝑆_{𝑛}$ be an automorphism of the symmetric group $𝑆_{𝑛}$ , and suppose $𝑛 \neq 6$ . Prove the following.

(a) $𝜑$ sends transpositions in $𝑆_{𝑛}$ to transpositions.

(b) $Aut (𝑆_{𝑛}) ≅ 𝑆_{𝑛}$ .

Problem 3.13

Let

𝐺

be a finite group and let

𝑓 : 𝐺 \to 𝐺

be a group homomorphism. Prove that there exists a positive integer

𝑁

such that

𝐺 = Ker (𝑓^{𝑁}) Im (𝑓^{𝑁})

and

Ker (𝑓^{𝑁}) \cap Im (𝑓^{𝑁}) = {𝑒}

Problem 3.14

Let $𝐺$ and $𝐻$ be finite groups. Solve the following problems.

(a) For every $𝑁 ⊴ 𝐺$ , if $𝐺 / 𝑁 ≅ 𝐻$ , prove that there are exactly $| Aut (𝐻) |$ surjective homomorphisms $𝑓 : 𝐺 ↠ 𝐻$ satisfying $Ker 𝑓 = 𝑁$ .

(b) Hence prove that $| {𝑓 : 𝐺 ↠ 𝐻 | 𝑓 is a group homomorphism} | = | Aut (𝐻) | | {𝑁 ⊴ 𝐺 | 𝐺 / 𝑁 ≅ 𝐻} |$ .

4 Direct Products, Free Products, and Semidirect Products

Problem 4.1

Let

𝑛 > 1

. If

𝑛

is odd, prove that

𝐷_{2 𝑛} ≅ 𝐷_{𝑛} \times ℤ_{2}

. Show that this conclusion fails when

𝑛

is even.

Problem 4.2

Determine whether each of the following statements is true, and explain your reasoning.

$𝐷_{𝑛} ≅ ℤ_{𝑛} \times ℤ_{2}$ .
$𝑆_{𝑛} ≅ 𝐴_{𝑛} \times ℤ_{2}$ .
${GL}_{2} (ℝ) ≅ {SL}_{2} (ℝ) \times ℝ^{*}$ .

Problem 4.3

Let

𝐺_{1}

and

𝐺_{2}

be finite groups such that

gcd (| 𝐺_{1} |, | 𝐺_{2} |) = 1

. Prove that

Aut (𝐺_{1} \times 𝐺_{2}) ≅ Aut (𝐺_{1}) \times Aut (𝐺_{2})

Problem 4.4

Solve the following problems.

(a) Give an example of $𝐻 < 𝐺_{1} \times 𝐺_{2}$ such that $𝐻$ cannot be written as the direct product of a subgroup $𝐻_{1} < 𝐺_{1}$ and a subgroup $𝐻_{2} < 𝐺_{2}$ .

(b) Let $𝐺_{1}$ and $𝐺_{2}$ be groups. Prove that every subgroup of $𝐺_{1} \times 𝐺_{2}$ has the form $𝐻_{1} \times 𝐻_{2}$ , where $𝐻_{𝑖} < 𝐺_{𝑖}$ for $𝑖 = 1, 2$ , if and only if for every $𝑔_{1} \in 𝐺_{1}$ and $𝑔_{2} \in 𝐺_{2}$ , both $𝑔_{1}$ and $𝑔_{2}$ have finite order and $gcd (𝑜 (𝑔_{1}), 𝑜 (𝑔_{2})) = 1$ .

Problem 4.5

Let $𝐺$ be a group satisfying $𝑥^{2} = 𝑒$ for every $𝑥 \in 𝐺$ .

(a) Prove that $𝐺$ is abelian.

(b) If $𝐺$ is finite, prove that there exists a nonnegative integer $𝑛$ such that $𝐺 ≅ \oplus_{𝑛} ℤ_{2}$ , the direct sum of $𝑛$ copies of $ℤ_{2}$ .

Problem 4.6

Let $𝐺_{1}$ and $𝐺_{2}$ be groups. Prove that

$𝐶 (𝐺_{1} \times 𝐺_{2}) = 𝐶 (𝐺_{1}) \times 𝐶 (𝐺_{2})$ ;
$[𝐺_{1} \times 𝐺_{2}, 𝐺_{1} \times 𝐺_{2}] = [𝐺_{1}, 𝐺_{1}] \times [𝐺_{2}, 𝐺_{2}]$ .

Problem 4.7

Solve the following problems.

(a) Let $𝐺 = 𝐺_{1} \times \dots \times 𝐺_{𝑛}$ be a direct product of groups, let $𝑝_{𝑖} : 𝐺 \to 𝐺_{𝑖}$ be the projection onto the $𝑖$ -th component, and let $𝜑_{𝑖} : 𝑆 \to 𝐺_{𝑖}$ be group homomorphisms for $𝑖 = 1, \dots, 𝑛$ . Prove that there exists a unique group homomorphism $𝜑 : 𝑆 \to 𝐺$ such that $𝑝_{𝑖} \circ 𝜑 = 𝜑_{𝑖}$ for every $𝑖 = 1, \dots, 𝑛$ .

(b) Let $𝐺 = 𝐺_{1} \times \dots \times 𝐺_{𝑛}$ , let $𝑗_{𝑖} : 𝐺_{𝑖} \to 𝐺$ be the natural homomorphisms, and let $𝜓_{𝑖} : 𝐺_{𝑖} \to 𝑆$ be group homomorphisms for $𝑖 = 1, \dots, 𝑛$ . Give an example showing that there need not exist a unique group homomorphism $𝜓 : 𝐺 \to 𝑆$ such that $𝜓 \circ 𝑗_{𝑖} = 𝜓_{𝑖}$ for every $𝑖 = 1, \dots, 𝑛$ .

Problem 4.8

Let $𝐺$ be a group and let $𝐻, 𝐾$ be subgroups of $𝐺$ . Suppose that

$𝐺 = 𝐻 𝐾$ ,
$𝐻 \cap 𝐾 = {𝑒}$ ,
$ℎ 𝑘 = 𝑘 ℎ$ for every $ℎ \in 𝐻$ and $𝑘 \in 𝐾$ .

Prove that the map $𝐻 \times 𝐾 \to 𝐺$ , $(ℎ, 𝑘) \mapsto ℎ 𝑘$ , is a group isomorphism, and that $𝐺 / 𝐻 ≅ 𝐾$ and $𝐺 / 𝐾 ≅ 𝐻$ .

Problem 4.9

Let ${𝐺_{𝛼}}_{𝛼 \in 𝐼}$ be a family of groups indexed by $𝐼$ . For a positive integer $𝑛$ , call an $𝑛$ -tuple $𝑎 = (𝑥_{1}, 𝑥_{2}, \dots, 𝑥_{𝑛})$ a word of length $𝑛$ over ${𝐺_{𝛼}}$ , where $𝑥_{𝑖} \in ⨆_{𝛼 \in 𝐼} 𝐺_{𝛼}$ (the symbol $⨆$ denotes disjoint union). Also define the empty word $𝑒$ of length $0$ .

(a) Define an equivalence relation $\sim$ on the set of all words over ${𝐺_{𝛼}}$ as follows.

(a1) If $𝑒_{𝛼}$ denotes the identity element of $𝐺_{𝛼}$ , then $(𝑥_{1}, \dots, 𝑥_{𝑖 - 1}, 𝑒_{𝛼}, 𝑥_{𝑖 + 1}, \dots, 𝑥_{𝑛}) \sim (𝑥_{1}, \dots, 𝑥_{𝑖 - 1}, 𝑥_{𝑖 + 1}, \dots, 𝑥_{𝑛})$ .

(a2) If $𝑥_{𝑖}$ and $𝑥_{𝑖 + 1}$ both belong to some $𝐺_{𝛼}$ , then $(𝑥_{1}, \dots, 𝑥_{𝑖}, 𝑥_{𝑖 + 1}, \dots, 𝑥_{𝑛}) \sim (𝑥_{1}, \dots, 𝑥_{𝑖} 𝑥_{𝑖 + 1}, \dots, 𝑥_{𝑛})$ .

Denote the equivalence class of a word $𝑎$ by $[𝑎]$ . For words $𝑎 = (𝑥_{1}, \dots, 𝑥_{𝑛})$ and $𝑏 = (𝑦_{1}, \dots, 𝑦_{𝑚})$ , define their product by $𝑎 𝑏 = (𝑥_{1}, \dots, 𝑥_{𝑛}, 𝑦_{1}, \dots, 𝑦_{𝑚})$ . Let $*_{𝛼 \in 𝐼} 𝐺_{𝛼}$ denote the set of all equivalence classes of words over ${𝐺_{𝛼}}$ . Prove that multiplication $[𝑎] [𝑏] = [𝑎 𝑏]$ is well-defined on $*_{𝛼 \in 𝐼} 𝐺_{𝛼}$ and makes it a group. This group is called the free product of ${𝐺_{𝛼}}_{𝛼 \in 𝐼}$ .

(b) Let $𝐺 = 𝐺_{1} * \dots * 𝐺_{𝑛}$ be a free product of groups, let $𝑞_{𝑖} : 𝐺_{𝑖} \to 𝐺$ send $𝑥_{𝑖}$ to $[(𝑥_{𝑖})]$ , and let $𝜓_{𝑖} : 𝐺_{𝑖} \to 𝑆$ be group homomorphisms for $𝑖 = 1, \dots, 𝑛$ . Prove that there exists a unique group homomorphism $𝜓 : 𝐺 \to 𝑆$ such that $𝜓 \circ 𝑞_{𝑖} = 𝜓_{𝑖}$ for every $𝑖 = 1, \dots, 𝑛$ .

Problem 4.10

Let $𝑆$ be a nonempty set. For $𝛼 \in 𝑆$ , let $𝐺_{𝛼} = {\dots, 𝛼^{- 1}, 𝑒_{𝛼} = 𝛼^{0}, 𝛼 = 𝛼^{1}, 𝛼^{2}, \dots}$ be the infinite cyclic group generated by $𝛼$ . The free product $*_{𝛼 \in 𝑆} 𝐺_{𝛼}$ is called the free group generated by $𝑆$ and is denoted by $𝐹 (𝑆)$ . Let $𝑅$ be a subset of $𝐹 (𝑆)$ , let $𝑁$ be the smallest normal subgroup of $𝐹 (𝑆)$ containing $𝑅$ , and put $𝐺 = 𝐹 (𝑆) / 𝑁$ . Then $𝐺$ is called the group generated by the set of generators $𝑆$ with the set of relations $𝑅$ , also written $𝐺 = ⟨ 𝑆 | 𝑅 ⟩$ . If $𝑆 = {𝑠_{1}, \dots, 𝑠_{𝑛}}$ and $𝑅 = {𝑟_{1}, \dots, 𝑟_{𝑚}}$ , this may be written in more detail as $⟨ 𝑠_{1}, \dots, 𝑠_{𝑛} | 𝑟_{1} = \dots = 𝑟_{𝑚} = 𝑒 ⟩$ . In particular, when $𝑟_{𝑖} = 𝛼^{- 𝑖} 𝛽^{𝑗}$ , the relation $𝑟_{𝑖} = 𝑒$ may also be written $𝛼^{𝑖} = 𝛽^{𝑗}$ .

(a) Determine the structure of the groups $⟨ 𝑟, 𝑠 | 𝑟^{𝑛} = 𝑠^{2} = 𝑒, 𝑠^{- 1} 𝑟 𝑠 = 𝑟^{- 1} ⟩$ and $⟨ 𝑖, 𝑗 | 𝑖^{4} = 𝑒, 𝑖^{2} = 𝑗^{2}, 𝑗^{- 1} 𝑖 𝑗 = 𝑖^{- 1} ⟩$ .

(b) Write down one possible set of generators and relations for $ℤ \times ℤ$ and for $𝑆_{4}$ .

Problem 4.11

Given groups $𝐻$ and $𝐾$ and a group homomorphism $𝜑 : 𝐾 \to Aut (𝐻)$ , one can define the semidirect product $𝐻 ⋊_{𝜑} 𝐾$ . As a set, $𝐻 ⋊_{𝜑} 𝐾 = {(ℎ, 𝑘) | ℎ \in 𝐻, 𝑘 \in 𝐾}$ , and the group operation is defined by $(ℎ, 𝑘) (ℎ^{'}, 𝑘^{'}) ≔ (ℎ 𝜑_{𝑘} (ℎ^{'}), 𝑘 𝑘^{'})$ , where $𝜑_{𝑘} ≔ 𝜑 (𝑘)$ , and the identity element is $(1_{𝐻}, 1_{𝐾})$ .

Verify that $𝐻 ⋊_{𝜑} 𝐾$ is a group.
Explain that the external direct product is a special case of the

semidirect product.

Verify the group isomorphisms $𝐻 ≅ 𝐻 \times 1_{𝐾} = {(ℎ, 1_{𝐾}) | ℎ \in 𝐻}, ℎ \mapsto (ℎ, 1_{𝐾}),$ $𝐾 ≅ 1_{𝐻} \times 𝐾 = {(1_{𝐻}, 𝑘) | 𝑘 \in 𝐾}, 𝑘 \mapsto (1_{𝐻}, 𝑘) .$ Also prove that $𝐻 ≅ 𝐻 \times 1_{𝐾}$ is a normal subgroup of $𝐻 ⋊_{𝜑} 𝐾$ .
Prove that $𝐾 ≅ 1_{𝐻} \times 𝐾$ is a normal subgroup of

$𝐻 ⋊_{𝜑} 𝐾$ if and only if $𝜑 : 𝐾 \to Aut (𝐻)$ is trivial, in which case $𝐻 ⋊_{𝜑} 𝐾 = 𝐻 \times 𝐾$ .

Let $𝐻$ and $𝐾$ be subgroups of a group $𝐺$ satisfying
- $𝐺 = 𝐻 𝐾$ ,
- $𝐻 \cap 𝐾 = {1}$ ,
- $𝐻 ⊴ 𝐺$ .
Consider the group homomorphism $𝜑 : 𝐾 \to Aut (𝐻)$ , $𝜑_{𝑘} (ℎ) = 𝑘 ℎ 𝑘^{- 1}$ for every $ℎ \in 𝐻$ . Prove that the map $𝑓 : 𝐻 ⋊_{𝜑} 𝐾 \to 𝐺$ , $(ℎ, 𝑘) \mapsto ℎ 𝑘$ , is a group isomorphism, and that $𝐺 / 𝐻 ≅ 𝐾$ .
Prove the following semidirect product decompositions.
- $𝐷_{𝑛} ≅ ⟨ 𝑟 ⟩ ⋊ {1, 𝑠} ≅ ℤ_{𝑛} ⋊ ℤ_{2}$ .
- $𝑆_{𝑛} ≅ 𝐴_{𝑛} ⋊ ⟨ 𝜏 ⟩ ≅ 𝐴_{𝑛} ⋊ ℤ_{2}$ , where $𝜏 \in 𝑆_{𝑛}$ is
any transposition.
- ${GL}_{2} (ℝ) ≅ {SL}_{2} (ℝ) ⋊ 𝐾 ≅ {SL}_{2} (ℝ) ⋊ ℝ^{*}$ , where
$𝐾 = {diag (𝑎, 1) | 𝑎 \in ℝ^{*}}$ .
For a fixed odd integer $𝑛 > 1$ , prove that the direct product

${SL}_{𝑛} (ℝ) \times ℝ^{*}$ can be realized as a nontrivial semidirect product ${SL}_{𝑛} (ℝ) ⋊ ℝ^{*}$ .

5 Group Actions and the Sylow Theorems

Problem 5.1

Let the group ${GL}_{3} (ℝ)$ act on the column-vector space $ℝ^{3}$ , regarded as a set, by matrix multiplication. Solve the following problems.

(a) Prove that the column vectors ${(1, 0, 0)}^{𝑡}$ and ${(0, 1, 0)}^{𝑡}$ lie in the same orbit.

(b) Is this group action transitive? Explain why.

Problem 5.2

Let

𝐺

be a group of order

𝑛

. Let

𝑝

be the smallest prime divisor of

𝑛

, and let

𝐻

be a subgroup of

𝐺

of index

𝑝

. Prove that

𝐻

is a normal subgroup of

𝐺

Problem 5.3

Let a group $𝐺$ act on a set $𝑋$ . The action of $𝐺$ on $𝑋$ is called faithful if, for every $𝑔 \in 𝐺$ satisfying $𝑔 𝑥 = 𝑥$ for all $𝑥 \in 𝑋$ , one has $𝑔 = 𝑒$ . It is called free if, for every $𝑥 \in 𝑋$ and $𝑔 \in 𝐺$ satisfying $𝑔 𝑥 = 𝑥$ , one always has $𝑔 = 𝑒$ . Determine whether each of the following group actions is faithful and whether it is free.

(a) The conjugation action of the dihedral group $𝐷_{4}$ on itself.

(b) The action of ${SO}_{𝑛} (ℝ)$ on $ℝ^{𝑛} ∖ {0}$ , where $𝑛 \geq 2$ .

Problem 5.4

Let $𝑈_{𝑛}$ denote the group of all $𝑛 \times 𝑛$ unitary matrices under matrix multiplication, and let $𝑆^{2 𝑛 - 1}$ denote the unit sphere in $ℂ^{𝑛}$ . Let $𝑈_{𝑛}$ act naturally on $𝑆^{2 𝑛 - 1}$ , where $𝑛 \geq 2$ .

(a) Prove that this action is transitive.

(b) Fix a point on $𝑆^{2 𝑛 - 1}$ . Prove that its stabilizer subgroup is isomorphic to $𝑈_{𝑛 - 1}$ .

Problem 5.5

Let $O_{𝑛} (ℝ) ≔ {𝑀 \in 𝑀_{𝑛} (ℝ) | 𝑀^{𝑡} 𝑀 = 𝐼}$ denote the set of all $𝑛 \times 𝑛$ real orthogonal matrices. It is a group under matrix multiplication and is called the orthogonal group of order $𝑛$ . Let ${SO}_{𝑛} (ℝ) ≔ {𝑀 \in O_{𝑛} (ℝ) | det 𝑀 = 1}$ . It is also a group under matrix multiplication and is called the special orthogonal group of order $𝑛$ .

(a) Prove that ${SO}_{𝑛} (ℝ) ⊴ O_{𝑛} (ℝ)$ and $[O_{𝑛} (ℝ) : {SO}_{𝑛} (ℝ)] = 2$ .

The group of all automorphisms of a regular $𝑛$ -hedron in $ℝ^{3}$ under composition is called the symmetry group of the regular $𝑛$ -hedron and is denoted by $𝐸_{𝑛}$ , where $𝑛 = 4, 6, 8, 12, 20$ . Regard $𝐸_{𝑛}$ as a subgroup of the orthogonal group $O_{3} (ℝ)$ . The group $𝐸_{𝑛} \cap {SO}_{3} (ℝ)$ is called the rotation group of the regular $𝑛$ -hedron and is denoted by $𝐹_{𝑛}$ , where $𝑛 = 4, 6, 8, 12, 20$ .

(b) For $𝑛 = 4, 6, 8, 12, 20$ , find the orders of $𝐸_{𝑛}$ and $𝐹_{𝑛}$ .

(d) Prove that $𝐹_{6} ≅ 𝐹_{8} ≅ 𝑆_{4}$ .

(e) Prove that $𝐹_{12} ≅ 𝐹_{20} ≅ 𝐴_{5}$ .

Problem 5.6

Prove that a group of order

56

and a group of order

𝑝 𝑞 𝑟

, where

𝑝 < 𝑞 < 𝑟

are primes, are not simple.

Problem 5.7

Let

𝐻

be a Sylow

𝑝

-subgroup of a finite group

𝐺

. Let

𝐿 < 𝐺

and suppose

𝑝 ∣ | 𝐿 |

. Prove that there exists

𝑔 \in 𝐺

such that

𝑔 𝐻 𝑔^{- 1} \cap 𝐿

is a Sylow

𝑝

-subgroup of

𝐿

Problem 5.8

Let

𝐺

be a group of order

𝑝^{2} 𝑞

, where

𝑝

and

𝑞

are primes and

𝑝 < 𝑞

. Prove that either

𝐺

has a normal Sylow

𝑞

-subgroup or

𝐺 ≅ 𝐴_{4}

Problem 5.9

Let

𝐺

be a simple group of order

60

. Prove that

𝐺

is isomorphic to

𝐴_{5}

. (Hint: You may assume that

𝐺

is not isomorphic to

𝐴_{5}

and prove that there exist two distinct Sylow

2

-subgroups

𝑃

and

𝑄

𝐺

satisfying

| 𝑁_{𝐺} (𝑃 \cap 𝑄) | \geq 12

, thereby obtaining a contradiction.)

Problem 5.10

Let $𝐺$ be a simple group of order $360$ . We will use proof by contradiction to determine the number of Sylow $3$ -subgroups of $𝐺$ .

(a) Assume that $𝐺$ has $40$ Sylow $3$ -subgroups. Prove that there exist two Sylow $3$ -subgroups $𝑃 \neq 𝑄$ such that $| 𝑃 \cap 𝑄 | > 1$ .

(b) Under the assumption in (a), let $𝐻 = 𝑃 \cap 𝑄$ . Prove that $𝐶_{𝐺} (𝐻) / 𝐻 ≅ 𝐴_{4}$ , and that $𝐶_{𝐺} (𝐻)$ has a normal Sylow $2$ -subgroup $𝐾$ , where $𝐶_{𝐺} (𝐻) = {𝑔 \in 𝐺 | \forall ℎ \in 𝐻, ℎ 𝑔 = 𝑔 ℎ}$ is the centralizer of $𝐻$ in $𝐺$ .

(d) Find the number of Sylow $3$ -subgroups of $𝐺$ .

6 Rings, Ideals, and Nilpotent Elements

Problem 6.1

Let $𝑅$ be a ring such that $𝑎^{2} = 𝑎$ for every $𝑎 \in 𝑅$ . Prove the following.

(a) For every $𝑎 \in 𝑅$ , $2 𝑎 = 𝑎 + 𝑎 = 0$ .

(b) $𝑅$ is commutative.

Problem 6.2

Solve the following problems.

(a) If $𝑅$ is a commutative ring and $𝑎, 𝑏 \in 𝑅$ are nilpotent, prove that $𝑎 + 𝑏$ is nilpotent.

(b) Give an example showing that, in a noncommutative ring, the sum of two nonzero nilpotent elements need not be nilpotent.

Problem 6.3

Let $𝐴 = ℤ [\sqrt{2}] = {𝑎 + 𝑏 \sqrt{2} | 𝑎, 𝑏 \in ℤ} \subset ℝ$ . For $𝑥 = 𝑎 + 𝑏 \sqrt{2} \in ℤ [\sqrt{2}]$ , define $𝑁 (𝑥) = 𝑎^{2} - 2 𝑏^{2}$ . Prove the following.

(a) $𝐴$ is a subring of the field $ℝ$ .

(b) $𝑥 \in {ℤ [\sqrt{2}]}^{*}$ if and only if $𝑁 (𝑥) = \pm 1$ .

Problem 6.4

Let $𝐹$ be a field and let $𝑇_{2} (𝐹)$ denote the set of all matrices $𝑇_{2} (𝐹) = {(\begin{matrix} 𝑎 & 𝑏 \\ 0 & 𝑐 \end{matrix}) : 𝑎, 𝑏, 𝑐 \in 𝐹}$ .

(a) Find all invertible elements of $𝑇_{2} (𝐹)$ .

(b) Prove that $𝐼 = {(\begin{matrix} 0 & 𝑏 \\ 0 & 0 \end{matrix}) : 𝑏 \in 𝐹}$ is an ideal satisfying $𝐼^{2} = 0$ , and prove that $𝑇_{2} (𝐹) / 𝐼 ≅ 𝐹 \times 𝐹$ .

Problem 6.5

Let $𝑅$ and $𝑆$ be unital rings. Prove that every ideal of $𝑅 \times 𝑆$ can be written uniquely in the form $𝐼 \times 𝐽$ , where $𝐼 ⊴ 𝑅$ and $𝐽 ⊴ 𝑆$ . Now suppose in addition that $𝑅$ and $𝑆$ are nonzero unital commutative rings. Prove the following.

(a) The prime ideals of $𝑅 \times 𝑆$ are exactly $𝑃 \times 𝑆$ and $𝑅 \times 𝑄$ , where $𝑃$ and $𝑄$ are prime ideals of $𝑅$ and $𝑆$ , respectively.

(b) The same conclusion about prime ideals also holds for maximal ideals.

Problem 6.6

Let

𝑛 \geq 2

be a positive integer and let

𝑅 = 𝑀_{𝑛} (ℂ)

be the ring of

𝑛 \times 𝑛

complex matrices. If

𝐼

is an ideal of

𝑅

, prove that

𝐼 = 0

𝐼 = 𝑅

Problem 6.7

Let

𝑅

be a commutative ring and let

𝐼 \subset 𝑅

be an ideal. Define

\sqrt{𝐼} = {𝑟 \in 𝑅 | there exists a positive integer 𝑛 such that 𝑟^{𝑛} \in 𝐼}

. Verify that

\sqrt{𝐼}

is an ideal.

Problem 6.8

Give an example of a maximal ideal

𝐼

in a noncommutative ring

𝑅

such that

𝐼

is not a prime ideal of

𝑅

Problem 6.9

Let $𝑅$ be a ring and let $Nil (𝑅) = {𝑎 \in 𝑅 | 𝑎 is nilpotent}$ .

(a) Suppose $𝑅$ satisfies the property that if $𝑎^{2} = 0$ , then $𝑎 = 0$ . Prove that $Nil (𝑅) = {0}$ .

(b) If $𝑅$ is commutative, prove that $Nil (𝑅)$ is an ideal.

Problem 6.10

Let

𝑅

be a unital ring. Let

{𝐼_{𝑖}}_{𝑖 \in Λ}

be a family of ideals of

𝑅

, where

Λ

is a nonempty index set. If for every

𝑖, 𝑗 \in Λ

one has

𝐼_{𝑖} \subseteq 𝐼_{𝑗}

𝐼_{𝑗} \subseteq 𝐼_{𝑖}

, prove that

⋃_{𝑖 \in Λ} 𝐼_{𝑖}

is an ideal of

𝑅

7 Ring Homomorphisms, Polynomial Rings, and Ideals

Problem 7.1

Let $𝑅$ be a unital commutative ring and let $𝐼_{1}, 𝐼_{2}, 𝐼_{3}$ be ideals of $𝑅$ .

(a) If $𝐼_{1}$ and $𝐼_{2}$ are comaximal, prove that $𝐼_{1} 𝐼_{2} = 𝐼_{1} \cap 𝐼_{2}$ .

(b) If $𝐼_{1}$ and $𝐼_{3}$ are comaximal, and $𝐼_{2}$ and $𝐼_{3}$ are comaximal, prove that $𝐼_{1} 𝐼_{2}$ and $𝐼_{3}$ are comaximal.

Problem 7.2

Let $𝑅$ and $𝑆$ be unital rings and let $𝑓 : 𝑅 \to 𝑆$ be a nonzero ring homomorphism. Solve the following problems.

(a) If $𝐼$ is a prime ideal of $𝑆$ , prove that $𝑓^{- 1} (𝐼)$ is a prime ideal of $𝑅$ .

(b) If $𝐽$ is a prime ideal of $𝑅$ , must $𝑓 (𝐽)$ be a prime ideal of $𝑆$ ? Explain why.

(d) If $𝐽$ is a maximal ideal of $𝑅$ , must $𝑓 (𝐽)$ be a maximal ideal of $𝑆$ ? Explain why.

Problem 7.3

Let $𝐴 = ℤ \times ℤ$ be an abelian group and define $End (𝐴) ≔ {𝑓 : 𝐴 \to 𝐴 | 𝑓 is a group homomorphism}$ .

(a) Prove that $End (𝐴)$ has a natural ring structure. State its addition and multiplication operations, and find the zero element and the multiplicative identity of $End (𝐴)$ .

(b) Prove that $End (𝐴)$ is isomorphic to the ring of $2 \times 2$ integer matrices.

Problem 7.4

Let $𝑓 : 𝑅 \to 𝑆$ be a ring homomorphism, let $𝐼$ and $𝐽$ be ideals of $𝑅$ and $𝑆$ , respectively, and suppose $𝑓 (𝐼) \subseteq 𝐽$ . Define $𝑓 : 𝑅 / 𝐼 \to 𝑆 / 𝐽$ , $𝑎 + 𝐼 \mapsto 𝑓 (𝑎) + 𝐽$ .

(a) Prove that $𝑓$ is well-defined and is a ring homomorphism.

(b) Prove that $𝑓$ is a ring isomorphism if and only if $𝑓 (𝑅) + 𝐽 = 𝑆$ and $𝑓^{- 1} (𝐽) = 𝐼$ .

Problem 7.5

Let $𝑅$ be a unital ring and let $𝑀_{𝑛} (𝑅)$ be the ring of $𝑛 \times 𝑛$ matrices over $𝑅$ . If $𝐼$ is an ideal of $𝑅$ , let $𝑀_{𝑛} (𝐼)$ denote the set of all $𝑛 \times 𝑛$ matrices whose entries all belong to $𝐼$ . Prove the following.

(a) $𝑀_{𝑛} (𝐼)$ is an ideal of $𝑀_{𝑛} (𝑅)$ .

(b) Every ideal of $𝑀_{𝑛} (𝑅)$ has the form $𝑀_{𝑛} (𝐽)$ , where $𝐽$ is some ideal of $𝑅$ .

Problem 7.6

Solve the following problems.

(a) Prove that $ℝ [𝑥] / (𝑥^{2} + 1)$ is isomorphic to the field of complex numbers $ℂ$ .

(b) Prove that the quotient ring $ℤ [𝑥] / (𝑥^{2} + 1)$ is isomorphic to the ring of Gaussian integers $ℤ [𝑖] = {𝑎 + 𝑏 𝑖 | 𝑎, 𝑏 \in ℤ}$ .

Problem 7.7

Let

𝑓 (𝑥) = 𝑥^{4} + 1

. Discuss separately whether

𝑓 (𝑥)

is irreducible in

ℚ [𝑥]

and whether it is irreducible in

ℝ [𝑥]

, and explain your reasoning.

Problem 7.8

Let $𝐴 = ℤ [\sqrt{2}] = {𝑎 + 𝑏 \sqrt{2} | 𝑎, 𝑏 \in ℤ}$ .

(a) Let $𝐼 = (𝑥^{2}) \subseteq 𝐴 [𝑥]$ be the ideal generated by $𝑥^{2}$ . Is the quotient ring $𝐴 [𝑥] / 𝐼$ an integral domain? Explain why.

(b) Is the ideal $𝐽 = (4, 𝑥)$ of $𝐴 [𝑥]$ principal? Explain why.

Problem 7.9

Let

𝐹

be a field. Prove that

𝐹 [𝑥]

contains infinitely many monic irreducible polynomials.

Problem 7.10

Let $𝑅$ be a unital commutative ring and let $𝑅 [𝑥]$ be the polynomial ring in one indeterminate over $𝑅$ . Let $𝑓 = 𝑎_{𝑛} 𝑥^{𝑛} + \dots + 𝑎_{1} 𝑥 + 𝑎_{0} \in 𝑅 [𝑥]$ , where $𝑎_{𝑖} \in 𝑅$ for $0 \leq 𝑖 \leq 𝑛$ and $𝑎_{𝑛} \neq 0$ . Prove the following.

(a) $𝑓$ is a unit in $𝑅 [𝑥]$ if and only if $𝑎_{0} \in 𝑅$ is a unit and $𝑎_{1}, \dots, 𝑎_{𝑛}$ are nilpotent elements of $𝑅$ .

(b) $𝑓$ is nilpotent in $𝑅 [𝑥]$ if and only if $𝑎_{0}, 𝑎_{1}, \dots, 𝑎_{𝑛}$ are all nilpotent.

Problem 7.11

Let

𝜑 : ℤ [𝑥] \to ℤ [𝑥]

be a nonzero ring homomorphism. Determine all such ring homomorphisms, and in each case determine when the homomorphism is injective, when it is surjective, and when it is a ring automorphism.

Problem 7.12

Let $𝑅$ be a unital commutative ring and let $𝑎_{1}, \dots, 𝑎_{𝑛} \in 𝑅$ , where $𝑛 \geq 1$ . Define the map $Φ : 𝑅 [𝑥] \to 𝑅^{𝑛}$ , $𝑓 (𝑥) \mapsto (𝑓 (𝑎_{1}), \dots, 𝑓 (𝑎_{𝑛}))$ . Solve the following problems.

(a) Prove that $Φ$ is surjective if and only if $𝑎_{𝑖} - 𝑎_{𝑗} \in 𝑅^{*}$ for every $𝑖 \neq 𝑗$ .

(b) Under the equivalent condition above, prove that $Ker Φ = (\prod_{𝑖 = 1}^{𝑛} (𝑥 - 𝑎_{𝑖}))$ , and hence prove that $𝑅 [𝑥] / (\prod_{𝑖 = 1}^{𝑛} (𝑥 - 𝑎_{𝑖})) ≅ 𝑅^{𝑛}$ .

(c) Under the equivalent condition above, prove that for every $𝑏_{1}, \dots, 𝑏_{𝑛} \in 𝑅$ , there exists a unique polynomial $𝑓 (𝑥) \in 𝑅 [𝑥]$ of degree less than $𝑛$ such that $𝑓 (𝑎_{𝑖}) = 𝑏_{𝑖}$ for all $1 \leq 𝑖 \leq 𝑛$ , and write down such a polynomial $𝑓 (𝑥)$ .

8 Domains, Fields of Fractions, and Factorization

Problem 8.1

Let

𝑅

be an integral domain and let

𝐹

be a field. Let

𝑖 : 𝑅 \to 𝐾 (𝑅)

𝑖 (𝑎) = \frac{𝑎}{1}

for all

𝑎 \in 𝑅

, be the natural embedding, and let

𝑓 : 𝑅 \to 𝐹

be an injective ring homomorphism. Prove that there exists a unique ring homomorphism

𝑓 : 𝐾 (𝑅) \to 𝐹

such that

𝑓 \circ 𝑖 = 𝑓

Problem 8.2

Let

𝑝

be a prime and let

𝐹

be a finite field of characteristic

𝑝

. Define a map

𝑓 : 𝐹 \to 𝐹

𝑓 (𝑎) = 𝑎^{𝑝}

for all

𝑎 \in 𝐹

. Prove that

𝑓

is a field isomorphism.

Problem 8.3

Let $𝑅$ be a unital commutative ring. Consider the ring homomorphism $𝑓 : ℤ \to 𝑅$ , $𝑓 (𝑛) = 𝑛 \cdot 1$ . If $Ker 𝑓 = 𝑚 ℤ$ , where $𝑚$ is a nonnegative integer, then $𝑚$ is called the characteristic of $𝑅$ .

(a) If $𝑅$ is an integral domain, prove that the characteristic of $𝑅$ is either a prime or $0$ .

(b) If $𝑅$ is a unital commutative ring of prime characteristic $𝑝$ , prove that for every $𝑎, 𝑏 \in 𝑅$ and every positive integer $𝑛$ , ${(𝑎 + 𝑏)}^{𝑝^{𝑛}} = 𝑎^{𝑝^{𝑛}} + 𝑏^{𝑝^{𝑛}}$ .

Problem 8.4

Let $𝑝$ be a prime and let $𝑥$ be an indeterminate. Put $𝐴 = {\sum_{𝑖 = 0}^{𝑛} 𝑎_{𝑖} 𝑥^{𝑝^{𝑖}} | 𝑛 \in ℤ_{\geq 0}, 𝑎_{𝑖} \in 𝔽_{𝑝}} \subseteq 𝔽_{𝑝 [𝑥]}$ . Use the usual polynomial addition on $𝐴$ , and define multiplication by $𝑓 * 𝑔 = 𝑓 \circ 𝑔 = 𝑓 (𝑔 (𝑥))$ . Solve the following problems.

(a) Prove that $𝐴$ is a unital commutative ring under the above addition and multiplication.

(b) Construct a ring isomorphism $𝔽_{𝑝 [𝑡]} ≅ 𝐴$ , where $𝑡$ is an indeterminate.

Problem 8.5

Let

(𝑅, 𝜑)

be a Euclidean domain that is not a field. Let

𝑅_{0}

be the set of all nonzero nonunit elements of

𝑅

, and let

𝑎 \in 𝑅_{0}

satisfy

𝜑 (𝑎) = {min}_{𝑏 \in 𝑅_{0}} 𝜑 (𝑏)

. Prove that the quotient ring

𝑅 / (𝑎) = {𝑟 + (𝑎) | 𝑟 \in 𝑅^{*} \cup {0}}

. In particular,

𝑅 / (𝑎)

is a field.

Problem 8.6

Give detailed proofs of Example 6 in this section.

(a) In the ring $ℤ [\sqrt{- 5}]$ , the ideal $(2, 1 + \sqrt{- 5})$ is not principal.

(b) In the ring $ℤ [𝑥]$ , the ideal $(2, 𝑥)$ is not principal.

Problem 8.7

Let $𝐹$ be a field. Let $𝐹 [[𝑥]] = {\sum_{𝑛 = 0}^{\infty} 𝑎_{𝑛} 𝑥^{𝑛} | 𝑎_{𝑛} \in 𝐹}$ be the set of all formal power series over $𝐹$ .

(a) For $𝑓 = \sum_{𝑛 = 0}^{\infty} 𝑎_{𝑛} 𝑥^{𝑛}$ and $𝑔 = \sum_{𝑛 = 0}^{\infty} 𝑏_{𝑛} 𝑥^{𝑛}$ in $𝐹 [[𝑥]]$ , define $𝑓 + 𝑔 = \sum_{𝑛 = 0}^{\infty} (𝑎_{𝑛} + 𝑏_{𝑛}) 𝑥^{𝑛}$ and $𝑓 𝑔 = \sum_{𝑛 = 0}^{\infty} (\sum_{𝑖 = 0}^{𝑛} 𝑎_{𝑖} 𝑏_{𝑛 - 𝑖}) 𝑥^{𝑛}$ . Verify that $𝐹 [[𝑥]]$ is a unital commutative ring under these addition and multiplication operations.

(b) Find the group of units of $𝐹 [[𝑥]]$ .

(c) Find all ideals of $𝐹 [[𝑥]]$ . Which of them are prime ideals and which are maximal ideals? Prove that $𝐹 [[𝑥]]$ is a PID.

(d) Find the field of fractions $𝐾 (𝐹 [[𝑥]])$ .

Problem 8.8

Let $𝑅 = ℤ [\frac{1 + \sqrt{- 19}}{2}] = {𝑎 + 𝑏 (\frac{1 + \sqrt{- 19}}{2}) | 𝑎, 𝑏 \in ℤ} \subset ℂ$ . Prove that $𝑅$ is not a Euclidean domain by completing the following steps.

(a) Assume that $𝑅$ is a Euclidean domain. Then there is a function $𝜑 : 𝑅 ∖ {0} \to ℕ_{0}$ satisfying the conditions in the definition of a Euclidean domain. Let $𝑅_{0}$ be the set of all nonzero nonunit elements of $𝑅$ , and let $𝑎 \in 𝑅_{0}$ satisfy $𝜑 (𝑎) = {min}_{𝑏 \in 𝑅_{0}} 𝜑 (𝑏)$ . Prove that $| 𝑅 / (𝑎) | \leq 3$ .

(b) Verify that $𝑓 (𝑥) = 𝑥^{2} - 𝑥 + 5 \in 𝑅 [𝑥]$ has a root in $𝑅$ , but that $𝑓 (𝑥) \in (𝑅 / (𝑎)) [𝑥]$ has no root in $𝑅 / (𝑎)$ . Derive a contradiction.

(c) Verify that $ℤ [\frac{1 + \sqrt{- 43}}{2}]$ , $ℤ [\frac{1 + \sqrt{- 67}}{2}]$ , and $ℤ [\frac{1 + \sqrt{- 163}}{2}]$ are all not Euclidean domains.

Problem 8.9

Let $𝐹$ be a field and let $𝑡$ be an indeterminate. Put $𝑅 = 𝐹 [𝑡^{2}, 𝑡^{3}] \subseteq 𝐹 [𝑡]$ . Solve the following problems.

(a) Prove that $𝐾 (𝑅) = 𝐹 (𝑡)$ .

(b) Let $𝐼 = (𝑡^{2}, 𝑡^{3})$ be the ideal of $𝑅$ . Prove that $𝐼$ is not principal, and hence prove that $𝑅$ is not a PID.

Problem 8.10

Let $𝐹$ be a field. Let $𝐹 [𝑥, 𝑦_{1}, 𝑦_{2}, \dots]$ be the polynomial ring over $𝐹$ in the indeterminates $𝑥, 𝑦_{1}, 𝑦_{2}, \dots$ , where each polynomial involves only finitely many indeterminates. Let $𝑅 = 𝐹 [𝑥, 𝑦_{1}, 𝑦_{2}, \dots] / (𝑥 - 𝑦_{1}^{2}, 𝑦_{1} - 𝑦_{2}^{2}, 𝑦_{2} - 𝑦_{3}^{2}, \dots)$ . Continue to denote the images of $𝑥, 𝑦_{1}, 𝑦_{2}, \dots$ in the quotient ring $𝑅$ by the same symbols. For every positive integer $𝑛$ , let $𝑅_{𝑛} = 𝐹 [𝑦_{𝑛}] \subseteq 𝑅$ . Solve the following problems.

(a) Prove that $𝑅_{1} \subseteq 𝑅_{2} \subseteq \dots$ , $𝑅 = ⋃_{𝑛 \geq 1} 𝑅_{𝑛}$ , and that every $𝑅_{𝑛}$ is isomorphic to the polynomial ring in one variable $𝐹 [𝑡]$ . Hence prove that every $𝑅_{𝑛}$ is a PID.

(b) Prove that every finitely generated ideal of $𝑅$ is principal.

(c) Let $𝐼 = (𝑦_{1}, 𝑦_{2}, 𝑦_{3}, \dots)$ . Prove that $𝐼$ is not finitely generated. Hence prove that $𝑅$ is not a PID.

Problem 8.11

Consider the subring $ℤ [\sqrt{- 3}] = {𝑎 + 𝑏 \sqrt{- 3} | 𝑎, 𝑏 \in ℤ}$ of the complex numbers. For $𝑥 = 𝑎 + 𝑏 \sqrt{- 3} \in ℤ [\sqrt{- 3}]$ , put $𝑁 (𝑥) = 𝑎^{2} + 3 𝑏^{2}$ .

(a) Prove that the multiplicative group ${ℤ [\sqrt{- 3}]}^{*} = {𝑥 \in ℤ [\sqrt{- 3}] | 𝑁 (𝑥) = 1} = {1, - 1}$ .

(b) Prove that $2$ , $1 + \sqrt{- 3}$ , and $1 - \sqrt{- 3}$ are pairwise nonassociate irreducible elements of $ℤ [\sqrt{- 3}]$ .

(d) Let $𝑥 = 4$ and $𝑦 = 2 (1 + \sqrt{- 3})$ . Prove that $𝑥$ and $𝑦$ have no greatest common divisor in $ℤ [\sqrt{- 3}]$ .

(e) Let $𝐼 = (2, 1 + \sqrt{- 3})$ . Prove that $𝐼$ is not a principal ideal of $ℤ [\sqrt{- 3}]$ .

Problem 8.12

Let $𝑅$ be an integral domain and let $𝑎, 𝑏 \in 𝑅$ be nonzero elements.

(a) If $𝑑 \in 𝑅$ is a greatest common divisor of $𝑎$ and $𝑏$ , and $𝑐 \in 𝑅$ is a least common multiple of $𝑎$ and $𝑏$ , prove that $𝑎 𝑏$ is associate to $𝑐 𝑑$ .

(b) If $lcm (𝑎, 𝑏)$ exists, prove that $gcd (𝑎, 𝑏)$ also exists.

Problem 8.13

Determine whether each of the following statements is true. Give a proof or a counterexample.

(a) If $𝐼$ is a nonzero prime ideal of $ℚ [𝑋]$ , then $ℚ [𝑋] / 𝐼$ is a unique factorization domain.

(b) If $𝐼$ is a nonzero prime ideal of $ℤ [𝑋]$ , then $ℤ [𝑋] / 𝐼$ is a unique factorization domain.

Problem 8.14

𝑅

is a UFD, prove that

𝑅

is a PID if and only if for any two principal ideals

(𝑎)

and

(𝑏)

𝑅

, their sum

(𝑎) + (𝑏)

is also principal.

9 Eisenstein’s Criterion, Modules, and Finite Abelian Groups

Problem 9.1

(Eisenstein’s criterion) Let

𝑅

be a UFD and let

𝐹

be the field of fractions of

𝑅

. Let

𝑓 (𝑥) \in 𝑅 [𝑥]

be a primitive polynomial and let

𝑝

be an irreducible element of

𝑅

. Write

𝑓 (𝑥) = 𝑎_{𝑛} 𝑥^{𝑛} + \dots + 𝑎_{1} 𝑥 + 𝑎_{0}

. If

𝑝! | 𝑎_{𝑛}

𝑝 | 𝑎_{𝑖}

for

0 \leq 𝑖 \leq 𝑛 - 1

, and

𝑝^{2}! | 𝑎_{0}

, prove that

𝑓 (𝑥)

is irreducible in

𝐹 [𝑥]

Problem 9.2

Let

𝑝

be an odd prime and let

𝑛 \geq 1

. Prove that the polynomial

𝑥^{𝑛} - 𝑝

is irreducible in

ℤ [𝑖] [𝑥]

Problem 9.3

Let $𝑅$ be an integral domain and let $𝑝 \in 𝑅 ∖ {0}$ . If the principal ideal $(𝑝)$ generated by $𝑝$ is a prime ideal, then $𝑝$ is called a prime element.

(a) Prove that $𝑝$ is a prime element if and only if, for every $𝑎, 𝑏 \in 𝑅$ , $𝑝 | 𝑎 𝑏$ implies $𝑝 | 𝑎$ or $𝑝 | 𝑏$ .

(b) Prove that every prime element is irreducible.

Problem 9.4

Let $𝐹$ be a field and let $𝑓 (𝑥, 𝑦, 𝑧) = 𝑥 𝑧 - 𝑦^{2} \in 𝐹 [𝑥, 𝑦, 𝑧]$ . Let $𝐼$ be the principal ideal generated by $𝑓$ .

(a) Prove that the quotient ring $𝐹 [𝑥, 𝑦, 𝑧] / 𝐼$ is an integral domain.

(b) Is $𝐹 [𝑥, 𝑦, 𝑧] / 𝐼$ a UFD? Explain why.

Problem 9.5

Let

𝐼 \subset ℤ [𝑥]

be a prime ideal. Prove that

𝐼

has at most two generators. Furthermore, find all prime ideals of

ℤ [𝑥]

Problem 9.6

Let

𝑅 = ℂ [𝑡]

, and let

𝑆

be a subring of

𝑅

that contains

ℂ

but is not equal to

ℂ

. Prove that

𝑅

is a finitely generated

𝑆

-module.

Problem 9.7

Let

𝑅

be a unital ring, let

𝑀

be a finitely generated

𝑅

-module, and let

𝑁

be a submodule of

𝑀

. Explain whether

𝑀 / 𝑁

and

𝑁

are also finitely generated.

Problem 9.8

Let

𝑅

be a PID. Prove that every submodule of a finitely generated free

𝑅

-module is again free.

Problem 9.9

Let

𝐾

be a field and let

𝐾 [𝑥]

be the polynomial ring in one indeterminate over

𝐾

. Let

𝐴 = {𝑀 | 𝑀 is a 𝐾 [𝑥] -module}

and

𝐵 = {(𝑉, 𝛼) | 𝑉 is a 𝐾 -vector space, 𝛼 \in {End}_{𝐾} (𝑉)}

. Prove that there is a natural bijection between

𝐴

and

𝐵

Problem 9.10

Let

𝐺

be a finite abelian group of order

100

. Find all isomorphism classes of

𝐺

Problem 9.11

Let

𝐺

be a finite abelian group of order

𝑛

. Let

𝑚

be a positive integer such that

𝑚 | 𝑛

. Prove that

𝐺

has a subgroup of order

𝑚

10 Field Extensions and Algebraic Elements

Problem 10.1

Let

(ℚ^{*}, \times)

be the multiplicative group of nonzero rational numbers. Is it a finitely generated abelian group? Is it a free abelian group? Explain why.

Problem 10.2

Let

𝐹

be a field, let

𝐹^{*} = 𝐹 ∖ {0}

be its multiplicative group, and let

𝐺

be a finite subgroup of

𝐹^{*}

. Prove that

𝐺

is cyclic.

Problem 10.3

Prove that there is no

3 \times 3

matrix

𝐴

over

ℚ

such that

𝐴^{8} = 𝐼

but

𝐴^{4} \neq 𝐼

. Here

𝐼

denotes the identity matrix.

Problem 10.4

Let

ℂ

be the field of complex numbers and let

𝑓 (𝑥_{1}, \dots, 𝑥_{𝑛}) \in ℂ [𝑥_{1}, \dots, 𝑥_{𝑛}]

, where

𝑛 \geq 2

and

deg 𝑓 \geq 1

. Prove that the zero set

𝑉 (𝑓)

𝑓 (𝑥_{1}, \dots, 𝑥_{𝑛})

is infinite.

Problem 10.5

Prove that

ℚ (\sqrt{2})

and

ℚ (\sqrt{5})

are isomorphic as

ℚ

-vector spaces, but are not isomorphic as fields.

Problem 10.6

Let

𝐿

be a finite extension of

𝐾

. Prove that the rational function field

𝐿 (𝑥)

is also a finite extension of

𝐾 (𝑥)

, and that

[𝐿 (𝑥) : 𝐾 (𝑥)] = [𝐿 : 𝐾]

Problem 10.7

Prove that the field of real numbers

ℝ

has no proper subfield isomorphic to itself. Does the field of complex numbers

ℂ

have a proper subfield isomorphic to itself?

Problem 10.8

Solve the following problems.

(a) Suppose $𝐾 / 𝐹$ is an algebraic extension of fields. Let $𝑅$ be a subring of $𝐾$ containing $𝐹$ . Prove that $𝑅$ is a subfield of $𝐾$ .

(b) Suppose $𝐾 / 𝐹$ is a field extension, and every subring $𝑅$ of $𝐾$ containing $𝐹$ is a field. Prove that $𝐾 / 𝐹$ is an algebraic extension.

Problem 10.9

Let

𝐸 / 𝐹

be a field extension of degree

𝑚

, and let

𝑓 (𝑥)

be an irreducible polynomial of degree

𝑛

over

𝐹

. If

𝑚

and

𝑛

are coprime, prove that

𝑓 (𝑥)

is irreducible over

𝐸

Problem 10.10

Let $𝛼 = \sqrt{2} + \sqrt{3}$ .

(a) Find the minimal polynomial of $𝛼$ over $ℚ$ .

(b) Prove that $ℚ (𝛼) = ℚ (\sqrt{2}, \sqrt{3})$ .

(c) Find the extension degree $[ℚ (𝛼) : ℚ]$ , and give a set of elements that are linearly independent over $ℚ$ and generate $ℚ (𝛼)$ .

Problem 10.11

Let

𝐹

be a field contained in the ring

𝑀_{𝑛} (ℚ)

𝑛 \times 𝑛

matrices over

ℚ

. Prove that

[𝐹 : ℚ] \leq 𝑛

11 Splitting Fields and Embeddings

Problem 11.1

For each of the following fields $𝐹$ and polynomials $𝑓 (𝑥) \in 𝐹 [𝑥]$ , find the splitting field of $𝑓$ over $𝐹$ .

(a) $𝐹 = ℚ$ , $𝑓 (𝑥) = 𝑥^{4} - 2$ .

(b) $𝐹 = ℚ$ , $𝑓 (𝑥) = 𝑥^{4} + 𝑥^{2} + 1$ .

Problem 11.2

Let

𝑝

be an odd prime and consider

𝑓 (𝑥) = 𝑥^{𝑝} - 2

. Let

𝐿

be the splitting field of

𝑓 (𝑥)

over

ℚ

. Prove that

[𝐿 : ℚ] = 𝑝 (𝑝 - 1)

Problem 11.3

The irreducibility of the cyclotomic polynomial $Φ_{𝑛} (𝑥)$ in $ℚ [𝑥]$ can be proved by the following steps.

(a) Let $𝑔 (𝑥) \in 𝔽_{𝑝 [𝑥]}$ and let $ℎ (𝑥) = 𝑔 (𝑥^{𝑝})$ . Prove that $ℎ (𝑥) = {𝑔 (𝑥)}^{𝑝}$ .

(b) Let $𝑓 (𝑥) \in ℤ [𝑥]$ be monic and irreducible, and suppose $𝑓 (𝑥) | Φ_{𝑛} (𝑥)$ . Let $𝜁 \in ℂ$ be a primitive $𝑛$ -th root of unity such that $𝑓 (𝜁) = 0$ . Let $𝑝$ be a prime relatively prime to $𝑛$ . Prove that $𝑓 (𝜁^{𝑝}) = 0$ . (Hint: Use the quotient map $ℤ [𝑥] \to 𝔽_{𝑝 [𝑥]}$ and the fact that $𝑥^{𝑛} - 1 \in 𝔽_{𝑝 [𝑥]}$ has no repeated roots in $𝔽_{𝑝}$ .)

Problem 11.4

Consider the polynomial $𝑓 (𝑥) = 𝑥^{4} - 4 𝑥^{2} + 2 \in ℚ [𝑥]$ and let $𝛼 = \sqrt{2 + \sqrt{2}}$ .

(a) Prove that $𝑓 (𝑥)$ is irreducible in $ℚ [𝑥]$ .

(b) Prove that $ℚ (𝛼)$ already contains every root of $𝑓 (𝑥)$ , and hence is the splitting field of $𝑓 (𝑥)$ .

Problem 11.5

Determine whether

ℂ (𝑥)

and

ℂ ((𝑥))

are algebraically closed fields, and give proofs. Here

ℂ (𝑥)

is the field of rational functions over the complex numbers, and

ℂ ((𝑥))

is the field of fractions of the formal power series ring

ℂ [[𝑥]]

Problem 11.6

Compute

Emb (ℚ (\sqrt[4]{2}, \sqrt{- 1}) / ℚ)

and

Emb (ℚ (\sqrt[3]{2}) / ℚ)

Problem 11.7

Give all field embeddings of

ℚ (\sqrt{2}, \sqrt{3})

into

ℂ

Problem 11.8

Prove that

ℚ (𝑥)

cannot be embedded into

ℚ

. What if

ℚ

is replaced by the field of complex numbers

ℂ

Problem 11.9

(a) Prove that every finite extension of $ℚ$ can be embedded into the field extension $ℂ / ℚ$ .

(b) Prove that every finitely generated extension of $ℚ$ can be embedded into the field extension $ℂ / ℚ$ .

Problem 11.10

Let $𝐹$ be a field and let $𝑓 (𝑥) \in 𝐹 [𝑥]$ be irreducible. Let $𝐾$ be the splitting field of $𝑓 (𝑥)$ over $𝐹$ , and suppose that there exists an element $𝛼 \in 𝐾$ such that both $𝛼$ and $𝛼 + 1$ are roots of $𝑓 (𝑥)$ .

(a) Prove that $char (𝐹) > 0$ .

(b) Prove that $char (𝐹)$ divides $| Aut (𝐾 / 𝐹) |$ .

12 Normal and Separable Extensions

Problem 12.1

For each of the following field extensions $𝐸 / 𝐹$ , determine whether it is normal and whether it is separable.

$𝐹 = ℚ$ , $𝐸 = ℚ (2^{\frac{1}{4}})$ .
$𝐹 = ℚ$ , $𝐸 = ℚ (\sqrt{2}, \sqrt{3}, \dots, \sqrt{𝑛}, \dots)$ .
$𝑝$ is a prime, $𝐹 = 𝔽_{𝑝 (𝑡^{𝑝})}$ , $𝐸 = 𝔽_{𝑝 (𝑡)}$ .
$𝐹 = ℚ (𝑥)$ , $𝐸 = ℚ (𝑥) [𝑡] / (𝑡^{2} - 𝑥)$ .

Problem 12.2

For a given prime

𝑝

, find the normal closure of the field extension

ℚ (2^{\frac{1}{𝑝}}) / ℚ

Problem 12.3

Give an example of field extensions

𝐹 \subseteq 𝐾 \subseteq 𝐿

such that both

𝐾 / 𝐹

and

𝐿 / 𝐾

are normal extensions, but

𝐿 / 𝐹

is not.

Problem 12.4

Let

𝐸 = ℂ (𝑡)

be the field of rational functions and let

𝐹 = ℂ (𝑡^{𝑛} + 𝑡^{- 𝑛})

. Prove that

𝐸 / 𝐹

is a normal extension.

Problem 12.5

For every prime

𝑝

, prove that

𝐸 = 𝔽_{𝑝 [𝑥]} / (𝑥^{𝑝} - 𝑥 + 1)

is a normal extension of

𝔽_{𝑝}

Problem 12.6

Let

𝐸 / ℚ

be a finite normal extension of odd degree. Prove that

𝐸 \subseteq ℝ

Problem 12.7

Let

𝐸 / 𝐹

be a field extension and let

𝛼, 𝛽 \in 𝐸

. If

𝐹 (𝛼) / 𝐹

is a normal extension and

𝐹 (𝛼) \cap 𝐹 (𝛽) = 𝐹

, prove that

[𝐹 (𝛼, 𝛽) : 𝐹] = [𝐹 (𝛼) : 𝐹] [𝐹 (𝛽) : 𝐹]

Problem 12.8

In $ℂ$ , let $𝛼 = \sqrt[3]{2}$ , let $𝜁_{3}$ be a primitive third root of unity, and put $𝐾 = ℚ (𝛼)$ and $𝑀 = ℚ (𝜁_{3} 𝛼)$ .

(a) Prove that both $𝐾 / ℚ$ and $𝑀 / ℚ$ are separable, nonnormal extensions of degree $3$ .

(b) Prove that $𝐾 \neq 𝑀$ , and hence prove that $𝐾 \cap 𝑀 = ℚ$ .

Problem 12.9

Let $𝑢$ be an indeterminate and let $𝐹 = ℚ (𝑢)$ . Let $𝛼^{3} = 𝑢$ and put $𝐸 = 𝐹 (𝛼)$ .

(a) Prove that $𝑥^{3} - 𝑢$ is irreducible in $𝐹 [𝑥]$ , and find $[𝐸 : 𝐹]$ .

(b) Prove that $𝐸 / 𝐹$ is separable but not normal.

(c) Let $𝜁_{3}$ be a primitive third root of unity. Prove that $𝐿 = 𝐹 (𝛼, 𝜁_{3})$ is the normal closure of $𝐸 / 𝐹$ , and find $[𝐿 : 𝐹]$ .

(d) Prove that $ℂ (𝑢, 𝛼) / ℂ (𝑢)$ is a normal extension.

Problem 12.10

Verify that the polynomial

𝑓 (𝑥) = 𝑥^{𝑝} - 𝑡

in Example 2 is irreducible in

𝐹 [𝑥]

13 Galois Theory and the Fundamental Theorem

Problem 13.1

Prove that the extension $𝐸 \supseteq 𝐹$ is Galois, determine the Galois group $Gal (𝐸 / 𝐹)$ , all of its subgroups, and the corresponding intermediate fields between $𝐹$ and $𝐸$ .

(a) $𝐹 = ℚ$ , $𝐸 = ℚ (\sqrt[4]{2}, 𝑖)$ .

(b) $𝐹 = ℝ (𝑥 + \frac{1}{𝑥})$ , $𝐸 = ℝ (𝑥)$ .

Problem 13.2

Compute the Galois group of

𝑥^{5} - 2

over

ℚ

and over

𝔽_{3}

, respectively.

Problem 13.3

Let

𝐸 / 𝐹

be a field extension, let

𝐺 = Aut (𝐸 / 𝐹)

, let

𝐻 < 𝐺

, and put

𝐾 = 𝐸^{𝐻}

. Prove that for every

𝜂 \in 𝐺

𝐸^{𝜂 𝐻 𝜂^{- 1}} = 𝜂 (𝐾)

Problem 13.4

𝐸 / ℚ

is a Galois extension and

Gal (𝐸 / ℚ) ≅ 𝑆_{5}

, prove that

𝐸

is the splitting field of a polynomial of degree

5

Problem 13.5

Let $𝐸 / 𝐹$ be a separable extension of degree $𝑑$ , and let $𝑁$ be the number of intermediate fields.

(a) Prove that $𝑁 \leq 2^{𝑑!}$ .

(b) Prove that $𝑁 \leq 2^{𝑑}$ .

Problem 13.6

Prove that every finite group can be realized as the Galois group of a Galois extension.

Problem 13.7

Suppose

𝐸 / 𝐹

is a Galois extension of degree

𝑝^{𝑛}

, where

𝑝

is a prime and

𝑛

is a positive integer. Prove that

𝐸

contains Galois extensions of

𝐹

of degree

𝑝

and of degree

𝑝^{𝑛 - 1}

Problem 13.8

Let

𝑓 (𝑥) \in 𝐹 [𝑥]

be an irreducible separable polynomial, and let

𝐸

be the splitting field of

𝑓 (𝑥)

. If

Gal (𝐸 / 𝐹)

is abelian, prove that

[𝐸 : 𝐹] = deg 𝑓 (𝑥)

Problem 13.9

Let $𝑝$ be a prime. Suppose that the degree of every finite extension of a field $𝐹$ is divisible by $𝑝$ .

(a) Prove that the degree of every finite separable extension of $𝐹$ is a power of $𝑝$ .

(b) Prove that the degree of every finite extension of $𝐹$ is a power of $𝑝$ .

Problem 13.10

Let $𝑘$ be a field, let $𝑛 \geq 2$ , and let $𝐸 = 𝑘 (𝑥_{1}, \dots, 𝑥_{𝑛})$ be the field of rational functions in $𝑛$ variables over $𝑘$ . Let $𝑆_{𝑛}$ act on $𝐸$ so that every $𝜎 \in 𝑆_{𝑛}$ fixes every element of $𝑘$ and satisfies $𝜎 (𝑥_{𝑖}) = 𝑥_{𝜎 (𝑖)}$ for $1 \leq 𝑖 \leq 𝑛$ . For $1 \leq 𝑟 \leq 𝑛$ , put $𝑒_{𝑟} = \sum_{1 \leq 𝑖_{1} < \dots < 𝑖_{𝑟} \leq 𝑛} 𝑥_{𝑖_{1}} \dots 𝑥_{𝑖_{𝑟}}$ , and let $𝐹 = 𝑘 (𝑒_{1}, \dots, 𝑒_{𝑛})$ . Solve the following problems.

(a) Prove that $𝐹 \subseteq 𝐸^{𝑆_{𝑛}}$ , and prove that $𝐸 / 𝐹$ is a finite extension with $[𝐸 : 𝐹] \leq 𝑛!$ .

(b) Prove that $𝐸 / 𝐹$ is a Galois extension, determine its Galois group, and thereby prove that $𝐸^{𝑆_{𝑛}} = 𝐹$ .

(c) Assume $char 𝑘 \neq 2$ and put $Δ = \prod_{1 \leq 𝑖 < 𝑗 \leq 𝑛} (𝑥_{𝑖} - 𝑥_{𝑗})$ . Prove that every element of $𝐴_{𝑛}$ fixes $Δ$ , while every odd permutation sends $Δ$ to $- Δ$ . Hence prove that $𝐸^{𝐴_{𝑛}} = 𝐹 (Δ)$ .

14 Möbius Inversion and Finite Fields

Problem 14.1

Define the Möbius function

𝜇 : ℕ \to ℤ

as follows:

𝜇 (𝑑) = {\begin{cases} 1 \\ if 𝑑 = 1 \\ {(- 1)}^{𝑘} \\ if 𝑑 = 𝑝_{1} \dots 𝑝_{𝑘} is the product of 𝑘 distinct primes \\ 0 \\ otherwise . \end{cases}

Let

𝑓, 𝑔 : ℕ \to ℕ

be nonzero functions. If

𝑓 (𝑛) = \sum_{𝑑 | 𝑛} 𝑔 (𝑑)

, prove that

𝑔 (𝑛) = \sum_{𝑑 | 𝑛} 𝜇 (𝑑) 𝑓 (\frac{𝑛}{𝑑})

, and use this to complete the calculation in Example 4.

Problem 14.2

Let $𝛼$ be a root in $𝔽_{3}$ of the polynomial $𝑥^{2} + 2 𝑥 + 2$ over $𝔽_{3}$ , and put $𝐸 = 𝔽_{3} [𝛼]$ . Find the following.

(a) The degree $𝑛 = [𝐸 : 𝔽_{3}]$ of the field extension.

(b) The inverse of $𝛽 = 𝛼^{2} + 1$ in $𝐸$ , expressed as an $𝔽_{3}$ -linear combination of $1, 𝛼, \dots, 𝛼^{𝑛 - 1}$ .

Problem 14.3

Find the splitting field of

𝑥^{8} - 1 \in 𝔽_{3} [𝑥]

Problem 14.4

Find the greatest common divisor of the polynomials

𝑓 (𝑥) = 𝑥^{6} + 𝑥^{4} + 𝑥 + 1 \in 𝔽_{2} [𝑥]

and

𝑔 (𝑥) = 𝑥^{8} - 𝑥 \in 𝔽_{2} [𝑥]

over

𝔽_{2}

. Use this to explain why

𝑓 (𝑥)

has exactly four roots in

𝔽_{8}

Problem 14.5

Find the order of the special linear group

{SL}_{𝑚} (𝔽_{𝑞})

over the finite field

𝔽_{𝑞}

Problem 14.6

Prove that every element of a finite field can be written as the sum of two squares.

Problem 14.7

For a prime

𝑝

and a positive integer

𝑛

, prove that

𝑥^{𝑝^{𝑛}} - 𝑥 + 1

is irreducible over

𝔽_{𝑝}

if and only if

𝑛 = 1

𝑛 = 𝑝 = 2

Problem 14.8

Let

𝑞

be a prime power, let

𝑓 (𝑥) \in 𝔽_{𝑞 [𝑥]}

be a monic irreducible polynomial of degree

𝑛

, and let

𝑚

be a positive integer. Prove that

𝑓 (𝑥)

factors in

𝔽_{𝑞^{𝑚}} [𝑥]

into exactly

gcd (𝑚, 𝑛)

pairwise distinct monic irreducible factors, and that each irreducible factor has degree

𝑛 / gcd (𝑚, 𝑛)

. In particular, prove that

𝑓 (𝑥)

is irreducible in

𝔽_{𝑞^{𝑚}} [𝑥]

if and only if

gcd (𝑚, 𝑛) = 1

Problem 14.9

Let $𝐾$ be a finite field and let $𝐾$ be an algebraic closure of $𝐾$ . Solve the following problems.

(a) Prove that $Aut (𝐾 / 𝐾)$ is an abelian group.

(b) Let $𝜎 \in Aut (𝐾 / 𝐾)$ . Prove that if there is a positive integer $𝑚$ such that $𝜎^{𝑚} = {id}_{𝐾}$ , then $𝜎 = {id}_{𝐾}$ . Hence prove that every nonidentity element of $Aut (𝐾 / 𝐾)$ has infinite order.

Problem 14.10

Let $𝐹$ be a field of characteristic $𝑝 > 0$ , let $𝑎 \in 𝐹$ , and suppose that $𝑓 (𝑥) = 𝑥^{𝑝} - 𝑥 - 𝑎 \in 𝐹 [𝑥]$ is irreducible. If $𝛼$ is a root of $𝑓 (𝑥)$ , put $𝐸 = 𝐹 (𝛼)$ . Solve the following problems.

(a) Prove that all roots of $𝑓 (𝑥)$ are exactly $𝛼 + 𝑐$ , where $𝑐 \in 𝔽_{𝑝}$ .

(b) Prove that $𝐸 / 𝐹$ is a Galois extension of degree $𝑝$ .

(c) For every $𝑐 \in 𝔽_{𝑝}$ , prove that there exists a unique $𝜎_{𝑐} \in Aut (𝐸 / 𝐹)$ such that $𝜎_{𝑐} (𝛼) = 𝛼 + 𝑐$ . Prove that $Aut (𝐸 / 𝐹) = {𝜎_{𝑐} | 𝑐 \in 𝔽_{𝑝}}$ , and hence prove that $Aut (𝐸 / 𝐹) ≅ (𝔽_{𝑝}, +)$ .

Problem 14.11

Let $𝑘$ be a finite field of characteristic $𝑝 > 0$ , let $𝐹 = 𝑘 (𝑡)$ be the field of rational functions in one variable over $𝑘$ , and let $𝐸$ be any finite extension of $𝐹$ . Prove the following.

(a) If $𝐸 / 𝐹$ is inseparable, then $𝑘 (𝑡^{\frac{1}{𝑝}}) \subseteq 𝐸$ .

(b) $𝐸 / 𝐹$ is a simple extension.