Abstract Algebra Exercise Solutions

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Problem 1.1

Prove that every subgroup of $\left(\mathbb{Z},+\right)$ is of the form

$$𝑚\mathbb{Z}=\left\{𝑚𝑡|𝑡\in \mathbb{Z}\right\},$$

where $𝑚\in {\mathbb{Z}}_{\ge 0}$.

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

1. Prove $𝐺\subseteq 𝑚\mathbb{Z}$. For any $𝑔\in 𝐺$, if $𝑔\notin 𝑚\mathbb{Z}$, write $𝑔=𝑘𝑚+𝑟$ where $𝑘\in \mathbb{Z}$, $𝑟\in \mathbb{N}$ and $0<𝑟<𝑚$. Then $𝑟=𝑔-𝑘𝑚\in 𝐺$, contradiction.
2. $𝑚\mathbb{Z}\subseteq 𝐺$ is obvious because $𝑚\in 𝐺$ and $𝑚\mathbb{Z}=⟨𝑚⟩\le 𝐺$.⁠$$∎
   

Problem 1.2

In a group $𝐺$, let ${𝐻}_1\le 𝐺$ and ${𝐻}_2\le 𝐺$, and define

$${𝐻}_1{𝐻}_2≔\left\{{ℎ}_1{ℎ}_2|{ℎ}_1\in {𝐻}_1,{ℎ}_2\in {𝐻}_2\right\}.$$

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, ${𝐻}_1=⟨\left(12\right)⟩=\left\{𝑒,\left(12\right)\right\}$, ${𝐻}_2=⟨\left(23\right)⟩=\left\{𝑒,\left(23\right)\right\}$.

${𝐻}_1{𝐻}_2=\left\{𝑒,\left(12\right),\left(23\right),\left(123\right)\right\}$ ²² Using the usual right-to-left composition, $\left(12\right)\left(23\right)=\left(123\right)$., but $\left(123\right)\left(12\right)=\left(13\right)\notin {𝐻}_1{𝐻}_2$.⁠$$∎

Problem 1.3

Let $𝐺$ be a finite group. Prove that for any two distinct elements $𝑎,𝑏\in 𝐺$, $𝑜\left(𝑎𝑏\right)=𝑜\left(𝑏𝑎\right)$.

Proposition 1.4

Let $𝐺$ be a group. Prove that for any two conjugate elements $𝑎,𝑏\in 𝐺$, $𝑜\left(𝑎𝑏\right)=𝑜\left(𝑏𝑎\right)$.

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

$${𝑏}^{𝑛}={\left(𝑔𝑎{𝑔}^{-1}\right)}^{𝑛}=𝑔{𝑎}^{𝑛}{𝑔}^{-1},$$

which implies

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

Problem 1.5

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 $ℎ\in 𝐺\backslash ⟨𝑔⟩$ of order $5$, then $ℎ,{ℎ}^2,{ℎ}^3,{ℎ}^4$ also are distinct elements of order 5. Observe that $\left\{𝑔,{𝑔}^2,{𝑔}^3,{𝑔}^4\right\}\cap \left\{ℎ,{ℎ}^2,{ℎ}^3,{ℎ}^4\right\}=\varnothing$. ³³ Otherwise one could deduce $ℎ\in \left\{𝑔,{𝑔}^2,{𝑔}^3,{𝑔}^4\right\}$.

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

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 ${ℝ}^{\ast }=ℝ\backslash \left\{0\right\}$. For any $𝑎\in {ℝ}^{\ast }$ and $𝑏\in ℝ$, define a map ${𝑓}_{𝑎,𝑏}:ℝ\to ℝ$ as follows:

$${𝑓}_{𝑎,𝑏}\left(𝑥\right)=𝑎𝑥+𝑏,\forall 𝑥\in ℝ.$$

Let $𝐺=\left\{{𝑓}_{𝑎,𝑏}|𝑎\in {ℝ}^{\ast },𝑏\in ℝ\right\}$ be the set of all such maps.

Problem 1.8

Let

$${\text{SL}}_2\left(\mathbb{Z}\right)=\left\{\left(\begin{array}{cc}𝑎& 𝑏\\ 𝑐& 𝑑\end{array}\right)|𝑎,𝑏,𝑐,𝑑\in \mathbb{Z},𝑎𝑑-𝑏𝑐=1\right\}.$$

Prove that ${\text{SL}}_2\left(\mathbb{Z}\right)$ is generated by the matrices

$$𝑻=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right),𝑺=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right).$$

Proof. Clearly, ${\text{SL}}_2\left(\mathbb{Z}\right)$ forms a group under matrix multiplication with identity ${𝑰}_{\mathrm{𝟐}}$. It suffices to prove that, by multiplying $𝑨=\left(\begin{array}{cc}𝑎& 𝑏\\ 𝑐& 𝑑\end{array}\right)\in {\text{SL}}_2\left(\mathbb{Z}\right)$ by finitely many $𝑻,𝑺,{𝑻}^{-1}$, we can always reduce it to ${𝑰}_2$.

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

When $𝑐\ne 0$, write $𝑎=𝑘𝑐+𝑟$ where $𝑘\in \mathbb{Z},0\le 𝑟<|𝑐|$. Do the following transformation

$$\left(\begin{array}{cc}𝑎& 𝑏\\ 𝑐& 𝑑\end{array}\right)\stackrel{{𝑻}^{-𝑘}}{\to }\left(\begin{array}{cc}𝑟& \ast \\ 𝑐& \ast \end{array}\right)\stackrel{𝑺}{\to }\left(\begin{array}{cc}𝑐& \ast \\ -𝑟& \ast \end{array}\right)$$

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 $\left(\begin{array}{cc}1& 𝑏\\ 0& 1\end{array}\right)$ or $\left(\begin{array}{cc}-1& 𝑏\\ 0& -1\end{array}\right)$ since $det𝑨=1$. Then we can further reduce it to ${𝑰}_2$:

$$\left(\begin{array}{cc}1& 𝑏\\ 0& 1\end{array}\right)\stackrel{{𝑻}^{-𝑏}}{\to }{𝑰}_2;$$

$$\left(\begin{array}{cc}-1& 𝑏\\ 0& -1\end{array}\right)\stackrel{{𝑻}^{𝑏}}{\to }-{𝑰}_2\stackrel{{𝑺}^2}{\to }{𝑰}_2.$$

⁠$$∎

Problem 1.9

Compute the cycle decomposition of $𝜎=\left(1357\right)\left(2345\right)\left(567\right)\in {𝑆}_7$.

Solution.

$$\begin{array}{c}1\to 1\to 1\to 3\\ 3\to 3\to 4\to 4\\ 4\to 4\to 5\to 7\\ 7\to 5\to 2\to 2\\ 2\to 2\to 3\to 5\\ 5\to 6\to 6\to 6\\ 6\to 7\to 7\to 1\end{array}$$

$\left(1357\right)\left(2345\right)\left(567\right)=\left(1347256\right)$.

Problem 1.10

Let $𝜎\in {𝑆}_{𝑛}\left(𝑛>2\right)$. 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 ${𝜎}_1,{𝜎}_2,\cdots ,{𝜎}_{𝑘}\left(𝑘\ge 2\right)$ be disjoint cycles in ${𝑆}_{𝑛}$, then $⟨{𝜎}_1⟩\cap ⟨{𝜎}_2,\dots ,{𝜎}_{𝑘}⟩=\left\{𝑒\right\}$.

Proof. Let

$$𝜎={𝜎}_1^{{𝑎}_1}=\prod _{𝑖=2}^{𝑘}{𝜎}_{𝑖}^{{𝑎}_{𝑖}},{𝑎}_{𝑖}\in \mathbb{Z}\text{for each}1\le 𝑖\le 𝑘.$$

We shall prove $𝜎=𝑒$. If ${𝜎}_1=𝑒$, we are done. Otherwise, take $𝑥\in \text{supp}\left({𝜎}_1\right)$, then $𝑥\notin \text{supp}\left({𝜎}_{𝑖}\right)$ for each $2\le 𝑖\le 𝑘$, which implies ${𝜎}_{𝑖}^{{𝑎}_{𝑖}}\left(𝑥\right)=𝑥$. Therefore

$$\begin{array}{cc}{𝜎}_1^{{𝑎}_1}\left(𝑥\right)& =\left(\prod _{𝑖=2}^{𝑘}{𝜎}_{𝑖}^{{𝑎}_{𝑖}}\right)\left(𝑥\right)\\ & =\left(\prod _{𝑖=2}^{𝑘-1}{𝜎}_{𝑖}^{{𝑎}_{𝑖}}\right)\left({𝜎}_{𝑘}^{{𝑎}_{𝑘}}\left(𝑥\right)\right)\\ & =\left(\prod _{𝑖=2}^{𝑘-1}{𝜎}_{𝑖}^{{𝑎}_{𝑖}}\right)\left(𝑥\right)\\ & =\cdots \\ & ={𝜎}_2^{{𝑎}_2}\left(𝑥\right)=𝑥.\end{array}$$

Let $𝑟$ be the length of ${𝜎}_1$, since ${𝜎}_1$ is a cycle, $𝑟\mid {𝑎}_1$. Note that $𝑜\left({𝜎}_1\right)=𝑟$, so $𝜎={𝜎}_1^{{𝑎}_1}=𝑒$.⁠$$∎

Proposition 1.12

Let ${𝜎}_1,{𝜎}_2,\cdots ,{𝜎}_{𝑘}$ 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 $𝑖$,

$$\begin{array}{cc}⟨{𝜎}_{𝑖}⟩\cap ⟨{𝜎}_1⟩\cdots ⟨{𝜎}_{𝑖-1}⟩⟨{𝜎}_{𝑖+1}⟩\cdots ⟨{𝜎}_{𝑘}⟩& =⟨{𝜎}_{𝑖}⟩\cap ⟨{𝜎}_1,\dots ,{𝜎}_{𝑖-1},{𝜎}_{𝑖+1},\dots ,{𝜎}_{𝑘}⟩\\ & =\left\{𝑒\right\}\end{array}$$

by Lemma 1.11.⁠$$∎

Problem 1.13

Let $𝑛>2$.

Problem 1.14

Find the number of permutations of order $6$ in ${𝑆}_8$.

Solution.

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 $𝑐\left(𝜎\right)$ be the sum of the number of these cycles and the number of fixed points of $𝜎$.

Problem 2.1

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

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

Prove that $\left[𝐴,𝐺\right]⊴𝐺$.

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

$$\begin{array}{cc}& {𝑔}^{\prime -1}{𝑎}^{-1}{𝑔}^{-1}𝑎𝑔{𝑔}^{\prime }\\ =& \left({𝑔}^{\prime -1}{𝑎}^{-1}{𝑔}^{\prime }\right)\left({𝑔}^{\prime -1}{𝑔}^{-1}{𝑔}^{\prime }\right)\left({𝑔}^{\prime -1}𝑎{𝑔}^{\prime }\right)\left({𝑔}^{\prime -1}𝑔{𝑔}^{\prime }\right)\\ =& {\left({𝑔}^{\prime -1}𝑎{𝑔}^{\prime }\right)}^{-1}{\left({𝑔}^{\prime -1}𝑔{𝑔}^{\prime }\right)}^{-1}\left({𝑔}^{\prime -1}𝑎{𝑔}^{\prime }\right)\left({𝑔}^{\prime -1}𝑔{𝑔}^{\prime }\right)\\ =& \left[{𝑔}^{\prime -1}𝑎{𝑔}^{\prime },{𝑔}^{\prime -1}𝑔{𝑔}^{\prime }\right]\in \left[𝐴,𝐺\right].\end{array}$$

Problem 2.2

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

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

⁠$$∎

Problem 2.3

Let $𝑆$ be a set of representatives of left cosets of a subgroup $𝐻\le 𝐺$, i.e. $𝐺={⨆}_{𝑎\in 𝑆}𝑎𝐻$. Prove that ${𝑆}^{-1}$ is a set of representatives of right cosets of $𝐻$, i.e. $𝐺={⨆}_{{𝑎}^{\prime }\in {𝑆}^{-1}}𝐻{𝑎}^{\prime }$.

Proof.

$$𝐺={𝐺}^{-1}=\underset{𝑎\in 𝑆}{⨆}{\left(𝑎𝐻\right)}^{-1}=\underset{𝑎\in 𝑆}{⨆}{𝐻}^{-1}{𝑎}^{-1}=\underset{𝑎\in 𝑆}{⨆}𝐻{𝑎}^{-1}=\underset{{𝑎}^{\prime }\in {𝑆}^{-1}}{⨆}𝐻{𝑎}^{\prime }.$$

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 $𝑎𝑁\in 𝐺/𝑁$ and $𝑜\left(𝑎𝑁\right)\mid \left|𝐺/𝑁\right|=𝑛$, so ${\left(𝑎𝑁\right)}^{𝑛}={𝑎}^{𝑛}𝑁=𝑁$, which implies ${𝑎}^{𝑛}\in 𝑁$.⁠$$∎

Problem 2.5

Let $𝐺$ be a finite group and let $𝐾⊴𝐻⊴𝐺$. If $gcd\left\{|𝐾|,|𝐻/𝐾|\right\}=1$, prove that $𝐾⊴𝐺$.

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

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

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

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

$$|𝐾\left({𝑔}^{-1}𝐾𝑔\right)|\mid \left|𝐻\right|\Rightarrow \frac{|𝐾\left({𝑔}^{-1}𝐾𝑔\right)|}{|𝐾|}\mid \frac{|𝐻|}{|𝐾|}.$$

On the other hand,

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

thus

$$\frac{|𝐾\left({𝑔}^{-1}𝐾𝑔\right)|}{|𝐾|}=\frac{|𝐾|}{|𝐾\cap {𝑔}^{-1}𝐾𝑔|}\Rightarrow \frac{|𝐾\left({𝑔}^{-1}𝐾𝑔\right)|}{|𝐾|}\mid \left|𝐾\right|.$$

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

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

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

Problem 2.6

Prove the following statements.

(a) For every positive integer $𝑛$, $⟨\frac{1}{𝑛}+\mathbb{Z}⟩$ is the unique subgroup of $\mathbb{Q}/\mathbb{Z}$ of order $𝑛$.

(b) Every subgroup of $\mathbb{Q}/\mathbb{Z}$ generated by finitely many elements is cyclic.

Problem 2.7

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

(a) $𝐻$ has $\left[𝐺:{𝑁}_{𝐺}\left(𝐻\right)\right]$ conjugates.

(b) The set of all conjugates of $𝐻$ cannot cover the whole group $𝐺$.

Problem 2.8

Let $𝐺$ be a finite group and let $𝐻,𝐾<𝐺$. If $gcd\left(\left[𝐺:𝐻\right],\left[𝐺:𝐾\right]\right)=1$, prove that $𝐺=𝐻𝐾$.

Problem 2.9

Let $𝐺$ be a subgroup of ${𝑆}_{2024}$, and suppose that $𝐺$ contains an $𝑛$-cycle, where $𝑛\ge 4$ is even. Prove that $𝐺$ is not simple.

Problem 2.10

Find the structure of each of the following two subgroups of ${𝑆}_5$.

(a) $⟨\left(12\right)\left(34\right),\left(12345\right)⟩$.

(b) $⟨\left(12\right)\left(35\right),\left(12345\right)⟩$.

Problem 2.11

For every positive integers $𝑛\ge 𝑚\ge 2$, regard ${𝐴}_{𝑚}$ naturally as a subgroup of ${𝐴}_{𝑛}$. Let ${𝐴}_{\infty }={\bigcup }_{𝑛\ge 2}{𝐴}_{𝑛}$.

(a) Explain that ${𝐴}_{\infty }$ has a natural group structure such that ${𝐴}_{𝑛}<{𝐴}_{\infty }$ for every $𝑛\ge 2$.

(b) Prove that ${𝐴}_{\infty }$ is simple.

Problem 2.12

Let $𝑛\ge 2$ be a positive integer.

(a) Let ${𝐺}_{𝑛}=\left[{𝑆}_{𝑛},{𝑆}_{𝑛}\right]$. Prove that ${𝐺}_{𝑛}={𝐴}_{𝑛}$ for $𝑛\ge 3$.

(b) Find $\left[{𝐴}_{𝑛},{𝐴}_{𝑛}\right]$.

Problem 3.1

Let ${𝐴}_4$ be the alternating group on four letters.

(a) Let $𝑁$ be a normal subgroup of ${𝐴}_4$ containing a $3$-cycle. Prove that $𝑁$ contains all $3$-cycles.

(b) Find all normal subgroups of ${𝐴}_4$, and explain your answer.

Problem 3.2

Describe all normal subgroups of ${𝑆}_{𝑛}$ for $𝑛\ge 1$.

Problem 3.3

For $𝑛\ge 2$, determine all integers $𝑖$ with $2\le 𝑖\le 𝑛$ such that ${𝑆}_{𝑛}=⟨\left(1𝑖\right),\left(12\cdots 𝑛\right)⟩$.

Problem 3.4

Let $𝑛$ be a positive integer. Prove that $𝜎,𝜏\in {𝑆}_{𝑛}$ are conjugate if and only if $𝜎$ and $𝜏$ have the same cycle structure.

Problem 3.5

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 $𝑓\left(𝑎\right)$ divides the order of $𝑎$.

Problem 3.6

Let $𝐺$ be a group and let $𝑓:𝐺\to 𝐺$ be the map defined by $𝑓\left(𝑥\right)={𝑥}^{-1}$ for all $𝑥\in 𝐺$.

(a) Give a necessary and sufficient condition for $𝑓$ to be a group homomorphism.

(b) If $𝐺$ is finite and $|\text{Aut}\left(𝐺\right)|=1$, prove that $|𝐺|\le 2$.

Problem 3.7

If a group $𝐺$ satisfies $𝐶\left(𝐺\right)=\left\{𝑒\right\}$, prove that $𝐶\left(\text{Aut}\left(𝐺\right)\right)=\{id\}$.

Problem 3.8

Let $𝐺$ be a finite group such that $|𝐺|={𝑝}_1{𝑝}_2\cdots {𝑝}_{𝑠}$, where the ${𝑝}_{𝑖}$ are distinct primes. If $𝐺$ has $𝑛$ normal subgroups, prove that $𝑛\le 2^{𝑠}$.

Problem 3.9

For $𝑛\ge 3$, find $|\text{Aut}\left({𝐷}_{𝑛}\right)|$.

Problem 3.10

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

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

(b) $\text{Aut}\left({𝑆}_{𝑛}\right)\cong {𝑆}_{𝑛}$.

Problem 3.11

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 $𝐺=\text{Ker}\left({𝑓}^{𝑁}\right)\text{Im}\left({𝑓}^{𝑁}\right)$ and $\text{Ker}\left({𝑓}^{𝑁}\right)\cap \text{Im}\left({𝑓}^{𝑁}\right)=\left\{𝑒\right\}$.

Problem 3.12

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

(a) For every $𝑁⊴𝐺$, if $𝐺/𝑁\cong 𝐻$, prove that there are exactly $|\text{Aut}\left(𝐻\right)|$ surjective homomorphisms $𝑓:𝐺\twoheadrightarrow 𝐻$ satisfying $\text{Ker}𝑓=𝑁$.

(b) Hence prove that $|\left\{𝑓:𝐺\twoheadrightarrow 𝐻|𝑓\text{is a group homomorphism}\right\}|=\left|\text{Aut}\left(𝐻\right)\right||\left\{𝑁⊴𝐺|𝐺/𝑁\cong 𝐻\right\}|$.

Problem 4.1

Let $𝑛>1$. If $𝑛$ is odd, prove that ${𝐷}_{2𝑛}\cong {𝐷}_{𝑛}\times {\mathbb{Z}}_2$. 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.

• ${𝐷}_{𝑛}\cong {\mathbb{Z}}_{𝑛}\times {\mathbb{Z}}_2$.

• ${𝑆}_{𝑛}\cong {𝐴}_{𝑛}\times {\mathbb{Z}}_2$.

• ${\text{GL}}_2\left(ℝ\right)\cong {\text{SL}}_2\left(ℝ\right)\times {ℝ}^{\ast }$.

Problem 4.3

Let ${𝐺}_1$ and ${𝐺}_2$ be finite groups such that $gcd\left(|{𝐺}_1|,|{𝐺}_2|\right)=1$. Prove that $\text{Aut}\left({𝐺}_1\times {𝐺}_2\right)\cong \text{Aut}\left({𝐺}_1\right)\times \text{Aut}\left({𝐺}_2\right)$.

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 $gcd\left(𝑜\left({𝑔}_1\right),𝑜\left({𝑔}_2\right)\right)=1$.

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 $𝐺\cong {\oplus }_{𝑛}{\mathbb{Z}}_2$, the direct sum of $𝑛$ copies of ${\mathbb{Z}}_2$.

Problem 4.6

Let ${𝐺}_1$ and ${𝐺}_2$ be groups. Prove that

• $𝐶\left({𝐺}_1\times {𝐺}_2\right)=𝐶\left({𝐺}_1\right)\times 𝐶\left({𝐺}_2\right)$;

• $\left[{𝐺}_1\times {𝐺}_2,{𝐺}_1\times {𝐺}_2\right]=\left[{𝐺}_1,{𝐺}_1\right]\times \left[{𝐺}_2,{𝐺}_2\right]$.

Problem 4.7

Solve the following problems.

(a) Let $𝐺={𝐺}_1\times \cdots \times {𝐺}_{𝑛}$ 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 $𝐺={𝐺}_1\times \cdots \times {𝐺}_{𝑛}$, 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

• $𝐺=𝐻𝐾$,

• $𝐻\cap 𝐾=\left\{𝑒\right\}$,

• $ℎ𝑘=𝑘ℎ$ for every $ℎ\in 𝐻$ and $𝑘\in 𝐾$.

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

Problem 4.9

Let ${\left\{{𝐺}_{𝛼}\right\}}_{𝛼\in 𝐼}$ be a family of groups indexed by $𝐼$. For a positive integer $𝑛$, call an $𝑛$-tuple $𝑎=\left({𝑥}_1,{𝑥}_2,\dots ,{𝑥}_{𝑛}\right)$ a word of length $𝑛$ over $\left\{{𝐺}_{𝛼}\right\}$, 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 $\left\{{𝐺}_{𝛼}\right\}$ as follows.

(a1) If ${𝑒}_{𝛼}$ denotes the identity element of ${𝐺}_{𝛼}$, then $\left({𝑥}_1,\dots ,{𝑥}_{𝑖-1},{𝑒}_{𝛼},{𝑥}_{𝑖+1},\dots ,{𝑥}_{𝑛}\right)\sim \left({𝑥}_1,\dots ,{𝑥}_{𝑖-1},{𝑥}_{𝑖+1},\dots ,{𝑥}_{𝑛}\right)$.

(a2) If ${𝑥}_{𝑖}$ and ${𝑥}_{𝑖+1}$ both belong to some ${𝐺}_{𝛼}$, then $\left({𝑥}_1,\dots ,{𝑥}_{𝑖},{𝑥}_{𝑖+1},\dots ,{𝑥}_{𝑛}\right)\sim \left({𝑥}_1,\dots ,{𝑥}_{𝑖}{𝑥}_{𝑖+1},\dots ,{𝑥}_{𝑛}\right)$.

Denote the equivalence class of a word $𝑎$ by $\left[𝑎\right]$. For words $𝑎=\left({𝑥}_1,\dots ,{𝑥}_{𝑛}\right)$ and $𝑏=\left({𝑦}_1,\dots ,{𝑦}_{𝑚}\right)$, define their product by $𝑎𝑏=\left({𝑥}_1,\dots ,{𝑥}_{𝑛},{𝑦}_1,\dots ,{𝑦}_{𝑚}\right)$. Let ${\ast }_{𝛼\in 𝐼}{𝐺}_{𝛼}$ denote the set of all equivalence classes of words over $\left\{{𝐺}_{𝛼}\right\}$. Prove that multiplication $\left[𝑎\right]\left[𝑏\right]=\left[𝑎𝑏\right]$ is well-defined on ${\ast }_{𝛼\in 𝐼}{𝐺}_{𝛼}$ and makes it a group. This group is called the free product of ${\left\{{𝐺}_{𝛼}\right\}}_{𝛼\in 𝐼}$.

(b) Let $𝐺={𝐺}_1\ast \cdots \ast {𝐺}_{𝑛}$ be a free product of groups, let ${𝑞}_{𝑖}:{𝐺}_{𝑖}\to 𝐺$ send ${𝑥}_{𝑖}$ to $\left[\left({𝑥}_{𝑖}\right)\right]$, 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 ${𝐺}_{𝛼}=\left\{\dots ,{𝛼}^{-1},{𝑒}_{𝛼}={𝛼}^0,𝛼={𝛼}^1,{𝛼}^2,\dots \right\}$ be the infinite cyclic group generated by $𝛼$. The free product ${\ast }_{𝛼\in 𝑆}{𝐺}_{𝛼}$ is called the free group generated by $𝑆$ and is denoted by $𝐹\left(𝑆\right)$. Let $𝑅$ be a subset of $𝐹\left(𝑆\right)$, let $𝑁$ be the smallest normal subgroup of $𝐹\left(𝑆\right)$ containing $𝑅$, and put $𝐺=𝐹\left(𝑆\right)/𝑁$. 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 $𝑆=\left\{{𝑠}_1,\dots ,{𝑠}_{𝑛}\right\}$ and $𝑅=\left\{{𝑟}_1,\dots ,{𝑟}_{𝑚}\right\}$, this may be written in more detail as $⟨{𝑠}_1,\dots ,{𝑠}_{𝑛}|{𝑟}_1=\cdots ={𝑟}_{𝑚}=𝑒⟩$. 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 $⟨𝑟,𝑠|{𝑟}^{𝑛}={𝑠}^2=𝑒,{𝑠}^{-1}𝑟𝑠={𝑟}^{-1}⟩$ and $⟨𝑖,𝑗|{𝑖}^4=𝑒,{𝑖}^2={𝑗}^2,{𝑗}^{-1}𝑖𝑗={𝑖}^{-1}⟩$.

(b) Write down one possible set of generators and relations for $\mathbb{Z}\times \mathbb{Z}$ and for ${𝑆}_4$.