Pool problems in group theory

Let $G$ be a group. Prove that $G$ is non-cyclic if and only if $G$ is the union of its proper subgroups.

Let $G$ be a group, and $G\times G$ the direct product. The set $D=\{(g,g,)\mid g\in G\}$ is a subgroup of $G\times G$. Prove that if $D$ is normal in $G\times G$ then $G$ is abelian.

The dihedral group, $D_8$, is the group of eight rigid symmetries of a square. Prove that $D_8$ is not the internal direct product of two of its proper subgroups.

Let $G$ be a finite group and $H,K\mathrel{\unlhd}G$ be normal subgroups of relatively prime order. Prove that $G$ is isomorphic to a subgroup of $G/H\times G/K$.

Suppose $G$ is a group that contains normal subgroups $H,K\unlhd G$ with $H\cap K=\{e\}$ and $HK=G$. Prove that $G\cong H\times K$.

Let $G$ be the group of upper-triangular real matrices $\begin{bmatrix} a & b \\ 0 & d\end{bmatrix}$ with $a,d\neq 0$, under matrix multiplication. Let $S$ be the subset of $G$ defined by $d=1$. Show that $S$ is normal and that $G/S\cong {\bf R}^{\times}$, the multiplicative group of nonzero real numbers.

Let $G$ be a group and suppose $\operatorname{Aut}(G)$ is trivial.
\begin{enumerate}[label=(\alph*)]
	\item Show that $G$ is abelian.
	\item Show that for any abelian group $H$, the {\bfseries inversion map} $\phi(h)=h^{-1}$ is an automorphism.
	\item Use parts (a) and (b) above to show that $g^2$ is the identity element for every $g\in G$.
\end{enumerate}

Let $G$ be a group and suppose $\operatorname{Aut}(G)$ is trivial.
\begin{enumerate}[label=(\alph*)]
	\item Show that $G$ is abelian.
	\item Show that for any abelian group $H$, the {\bfseries inversion map} $\phi(h)=h^{-1}$ is an automorphism.
	\item Use parts (a) and (b) above to show that $g^2$ is the identity element for every $g\in G$.
\end{enumerate}

Let $G$ be a group and $a\in G$ be an element. Let $n\in {\bf N}$ be the smallest positive number such that $a^n=e$, where $e$ is the identity element. Show that the set
\[
	\{e,a,a^2,\ldots, a^{n-1}\}
\]
contains no repetitions.

Let $G$ be a finite abelian group of odd order. Let $\phi:G\to G$ be the function defined by $\phi(g)=g^2$ for all $g\in G$. Prove that $\phi$ is an automorphism.

 Let $G$ be a group with exactly two conjugacy classes. Prove that $G$ is abelian, and describe all such groups (with proof).

Let ${\bf Z}_n$ denote the cyclic group of order $n$. Suppose $m\in {\bf N}$ is relatively prime to $n$. Define the function $\mu_m:{\bf Z}_n\to {\bf Z}_n$ by $m[a]_n=[ma]_n$.
\begin{enumerate}[label=\alph*)]
	\item Prove that the map $\mu_m$ is a well-defined automorphism of ${\bf Z}_n$.
	\item Prove that any automorphism of ${\bf Z}_n$ has the form $\mu_m$ for some $m$.
\end{enumerate}

For a group $G$ and an element $g\in G$, the {\bfseries centralizer} of $g$ in $G$ is the subgroup
\[
	C_G(g)=\{h\in G:hgh^{-1}=g\}.
\]
We say $g$ and $g’$ are {\bfseries conjugate in $G$} if there exists an element $h\in G$ such that $g’=hgh^{-1}$.

Suppose $S_n$ is a symmetric group with $n\geq 4$, and $\sigma$ is one of the $(n-2)$-cycles in $S_n$. (There are $\frac{n!}{2(n-2)}$ such cycles.)
\begin{enumerate}[label=\alph*)]
	\item Prove that $[S_n:C_{S_n}(\sigma)]=[A_n:C_{A_n}(\sigma)]$.
	\item Determine whether all $(n-2)$-cycles are conjugate in $A_n$.
\end{enumerate}

Let $G$ be a finite group and $n>1$ an integer such that $(ab)^n=a^n b^n$ for all $a,b\in G$. Let
 \[
	 G_n=\{c\in G\mid c^n=e\}\qquad\text{and}\qquad G^n=\{c^n\mid c\in G\}
\]
You may take for granted that these are subgroups. Prove that both $G_n$ and $G^n$ are normal in $G$, and $|G^n|=[G:G_n]$.

Suppose $G$ is a group, $H\leq G$ a subgroup, and $a,b\in G$. Prove that the following are equivalent:
\begin{enumerate}[label=\alph*)]
	\item $aH=bH$
	\item $b\in aH$
	\item $b^{-1}a\in H$
\end{enumerate}

Let $G$ be a group, and let $\operatorname{Aut}(G)$ denote the group of automorphisms of $G$. There is a homomorphism $\gamma:G\to \operatorname{Aut}(G)$ that takes $s\in G$ to the automorphism $\gamma_s$ defined by $\gamma_s(t)=sts^{-1}$.
	\begin{enumerate}[label=\alph*)]
		\item Prove rigorously, possibly with induction, that is $\gamma_s(t)=t^b$, then $\gamma_{s^n}(t)=t^{b^n}$.
		\item Suppose $s\in G$ has order 5, and $sts^{-1}=t^2$. Find the order of $t$. Justify your answer.
	\end{enumerate}

Let $G$ be an abelian group and $G_T$ be the set of elements of finite order in $G$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Prove that $G_T$ is a subgroup of $G$.
	\item Prove that every non-identity element of $G/G_T$ has infinite order.
	\item Characterize the elements of $G_T$ when $G={\bf R}/{\bf Z}$, where ${\bf R}$ is the additive group of real numbers.
\end{enumerate}

Suppose $G$ is a finite group of even order.
\begin{enumerate}[label=\alph*)]
	\item Prove that an element in $G$ has order dividing 2 if and only if it is its own inverse.
	\item Prove that the number of elements in $G$ of order 2 is odd.
	\item Use (b) to show $G$ must contain a subgroup of order 2.
\end{enumerate}

Let $N$ be a finite normal subgroup of $G$. Prove there is a normal subgroup $M$ of $G$ such that $[G:M]$ is finite and $nm=mn$ for all $n\in N$ and $m\in M$.

\medskip
\noindent ({\itshape Hint:} You may use the fact that the centralizer $C(h):=\{g\in G\mid ghg^{-1}=h\}$ is a subgroup of $G$.)

Show that every finite group with more than two elements has a nontrivial automorphism.

Suppose $G_1$ and $G_2$ are groups, with identity elements $e_1$ and $e_2$, respectively. Prove that if $\phi:G_1\to G_2$ is an isomorphism, then $\phi(e_1)=e_2$.

Suppose $A$ and $B$ are subgroups of a group $G$, and suppose $B$ is of finite index in $G$.
\begin{enumerate}[topsep=0.1in]
	\item Show that the index of $A\cap B\leq A$ is finite, and in fact $|A:A\cap B|\leq |G:B|$. {\itshape Hint:} Find a set map $A/A\cap B\to G/B$.
	\item Prove that equality holds in (a) if and only if $G=AB$.
\end{enumerate}

Let $G$ be a group. For each $a\in G$, let $\gamma_a$ denote the automorphism of $G$ defined by $\gamma_a(b)=aba^{-1}$ for all $b\in G$. The set $\operatorname{Inn}(G)=\{\gamma_a:a\in G\}$ is a subgroup of the automorphism group of $G$, called the subgroup of {\bfseries inner automorphisms}.

\medskip
Prove that $\operatorname{Inn}(G)$ is isomorphic to $G/Z(G)$, where $Z(G)$ is the center of $G$.

The additive group ${\bf Z}=({\bf Z},+)$ of rational integers is a subgroup of the additive group ${\bf Q}=({\bf Q},+)$. Show that ${\bf Z}$ has infinite index in ${\bf Q}$.

Let $G$ be a group of order $2p$, where $p$ is an odd prime. Prove $G$ contains a nontrivial, proper normal subgroup.

Prove from the definition along that there are no nonabelian groups of order less than $5$.

Let $G$ be a group and $H,K\mathrel{\unlhd}G$ be normal subgroups with $H\cap K=\{e\}$. Show that each element in $H$ commutes with every element in $K$.

Let $G$ be a group and $N$ a normal subgroup of $G$. Let $aN$ denote the left coset defined by $a\in G$, and consider the binary operation
\[
	G/N\times G/N\to G/N
\]
given by $(aN, bN)\mapsto abN$.
\begin{enumerate}[label=\alph*)]
	\item Show the operation is well defined.
	\item Show the operation is well defined only if the subgroup $N$ is normal.
\end{enumerate}

Let $H$ be a subgroup of a group $G$. The {\bfseries normalizer} of $H$ in $G$ is the set ${\bf N}_G(H)=\{g\in G\,\mid\, gH=Hg\}$.
\begin{enumerate}[label=\alph*)]
	\item Prove ${\bf N}_G(H)$ is a subgroup of $G$ containing $H$.
	\item Prove ${\bf N}_G(H)$ is the largest subgroup of $G$ in which $H$ is normal.
\end{enumerate}

Let $G$ be a group and suppose $H\leq G$. The {\bfseries normalizer} of $H$ in $G$ is defined to be $N(H)=\{g\in G\,|\, gH=Hg\}$ and the {\bfseries centralizer} of $H$ in $G$ is defined to be $C(H)=\{g\in G\,|\,  gh=hg\text{ for all }h\in H\}$.
\begin{enumerate}[label=(\alph*)]
	\item Prove that $N(H)$ is a subgroup of $G$.
	\item Prove that $C(H)$ is a normal subgroup of $N(H)$ and that $N(H)/C(H)$ is isomorphic to a subgroup of $\operatorname{Aut}(H)$.
\end{enumerate}

Suppose $G$ is a cyclic group of order $n$, and $t\in G$ is a generator.
\begin{enumerate}[label=\alph*)]
	\item Give a positive integer $d$ such that $t^{-1}=t^d$.
	\item Let $c$ be an integer and let $m=\gcd(n,c)$. Prove that the order of $t^c$ is $\frac{n}{m}$.
\end{enumerate}

Let $G$ be a finite group. Prove {\itshape from the definitions} that there exists a number $N$ such that $a^N=e$ for all $a\in G$.

Suppose $G$ is a group and $N\unlhd G$ is a finite normal subgroup. Prove that if $G/N$ contains an element of order $n$, then $G$ also contains an element of order $n$.

Suppose $\phi:G\to G'$ is a surjective homomorphism, $H\leq G$ is a subgroup containing $\ker(\phi)$, and $H'=\phi(H)$. Prove $\phi^{-1}(H')=H$, where $\phi^{-1}(H')=\{g\in G\,\mid\, \phi(g)\in H'\}$. Make sure to state explicitly where each hypothesis is used.

Let $G$ be a group, and $H, K$ be subgroups of $G$. Let $HK=\{hk\,\mid \, h\in H, k\in K\}$ denote the set product. Prove that $HK$ is a group if and only if $HK=KH$.

Suppose $G$ is a nontrivial finite group and $H,K\mathrel{\unlhd}G$ are normal subgroups with $\gcd(|H|,|K|)=1$.
\begin{enumerate}[label=\alph*)]
	\item Define a nontrivial group homomorphism $\phi:G\to G/H\times G/K$
	\item Prove $G$ is isomorphic to a subgroup of $G/H\times G/K$.
	\item Suppose $\gcd(m,n)=1$. Prove ${\bf Z}_{mn}\cong {\bf Z}_m\times {\bf Z}_n$.
\end{enumerate}

Suppose $G$ is a group, $H$ and $K$ are normal subgroups of $G$, and $H\leq K$.
\begin{enumerate}[label=\alph*)]
	\item Define a group homomorphism from $K$ to $G/H$.
	\item Compute the kernel of the homomorphism in (a), and apply the First Isomorphism Theorem.
\end{enumerate}

Let $G$ be a finite group and $\operatorname{Z}(G)$ denote its center.
\begin{enumerate}[label=\alph*)]
	\item Prove that if $G/\operatorname{Z}(G)$ is cyclic, then $G$ is abelian.
	\item Prove that if $G$ is nonabelian, then $|\operatorname{Z}(G)|\leq \frac{1}{4}|G|$.
\end{enumerate}

Let $G$ be a group, $m\in {\bf N}$, and $g\in G$ be an element such that $g^m=e$. Prove that $\operatorname{o}(g)\mid m$, where $\operatorname{o}(g)$ is the order of $g$.

\begin{enumerate}[label=(\alph*)]
	\item Show that if $G$ is any group (not necessarily finite) and $H$ is a subgroup, then $G$ is a disjoint union of left cosets of $H$.
	\item State and prove Lagrange's Theorem for finite groups.
\end{enumerate}

Let $G$ be a group and $H\leq G$ a subgroup. For each coset $aH$ of $H$ in $G$, define the set
\[
	G_{aH}=\{b\in G\,|\,baH=aH\}.
\]
\begin{enumerate}[label=\alph*)]
	\item Prove that $G_{aH}$ is a subgroup of $G$.
	\item Suppose that $H$ is normal in $G$. Prove that $G_{aH}=H$.
\end{enumerate}

Let $G$ be a group of order $2n$ for some positive integer $n > 1$.
\begin{enumerate}[label=\alph*)]
	\item Prove there exists a subgroup $K$ of $G$ of order $2$.
	\item Suppose $K$ in (a) is a \underline{normal} subgroup. Prove that $K$ is contained in the center $\operatorname{Z}(G)$. (Recall $\operatorname{Z}(G)=\{a\in G\mid ab=ba\text{ for all }b\in G\}$.)
\end{enumerate}

\begin{enumerate}[label=\alph*)]
	\item Suppose $N$ is a normal subgroup of a group $G$ and $\pi_N:G\to G/N$ is the usual projection homomorphism, defined by $\pi_N(g)=gN$. Prove that if $\phi:G\to H$ is any homomorphism with $N\leq \ker(\phi)$, then there exists a unique homomorphism $\psi:G/N\to H$ such that $\phi = \psi\circ \pi_N$. (You must explicitly define $\psi$, show it is well defined, show $\phi=\psi\circ\pi_N$, and show that $\psi$ is uniquely determined.)
	\item Prove the:
		\medskip
		{\bfseries Third Isomorphism Theorem.} If $M, N\unlhd G$ with $N\leq M$, then $(G/N)/(M/N)\cong G/M$.
\end{enumerate}