Template problems in group theory

Let $G$ and $H$ be groups of order 10 and 15, respectively. Prove that if there is a nontrivial homomorphism $\phi:G\to H$, then $G$ is abelian.

Let $n$ be a number between $0$ and $10$. Compute $n^{111}\pmod{11}$, expressing your answer as a number between $0$ and $10$. Give as detailed a proof as you can, justifying every step, no matter who trivial you think it is.

Let $S_n$ denote the symmetric group on $n$ letters.
\begin{enumerate}[label=\alph*)]
	\item Is the element $(1\,2\,3\,4)(2\,5\,3\,4\,6)(1\,5\,3\,2\,4\,7)\in S_7$ even or odd? Indicate your reasoning.
	\item Find the order of $(1\,3\,4)(2\,4\,3)(1\,3\,4)\in S_4$. Show all work.
	\item Write $(1\,5\,2\,3)(2\,1\,3\,4)(1\,5\,2\,3)^{-1}\in S_5$ in disjoint cycle form. Show all work.
\end{enumerate}

Determine with proof the automorphism group $\operatorname{Aut}(V)$ of the Klein 4-group $V=\{e,a,b,ab\}$. To what familiar group is it isomorphic?

Determine the number of group homomorphisms $\phi$ between the given groups. Here $K_4$ denotes the Klein four-group (also known as ${\bf Z}/2{\bf Z}\times {\bf Z}/2{\bf Z}$) and $S_3$ denotes the symmetric group on three elements.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item $\phi:K_4\to {\bf Z}/2{\bf Z}$
	\item $\phi:{\bf Z}/2{\bf Z}\to K_4$
	\item $\phi:S_3\to K_4$
	\item $\phi:K_4\to S_3$
\end{enumerate}

Without using Cauchy's Theorem or the Sylow theorems, prove that every group of order 21 contains an element of order three.

Explicitly list all group homomorphisms $f: {\bf Z}/6{\bf Z} \to {\bf Z}/12{\bf Z}$.

Let $C$ be a (possibly infinite) cyclic group, and let $\operatorname{Aut}(C)$ and $\operatorname{Inn}(C)$ be the groups of automorphisms and inner automorphisms, respectively. (Recall an automorphism $\gamma$ is {\bfseries inner} if it is given by conjugation: $\gamma(b)=aba^{-1}$ for some $a\in C$.)
\begin{enumerate}[label=\alph*)]
	\item Describe $\operatorname{Aut}(C)$ and $\operatorname{Inn(C)}$ in familiar terms, as groups you would study in a first algebra course. Prove your result. ({\itshape Hint:} Where do generators go?)
	\item Write $\operatorname{Aut}({\bf Z}_{12})$ down explicitly, giving its generic name and computing the order of every element. Show all work.
\end{enumerate}

Let $A_5$ denote the alternating group on a $5$-element set $\{1,2,3,4,5\}$. The set of automorphisms of $A_5$ form a group, denoted $\operatorname{Aut}(A_5)$. The group of {\bfseries conjugations} of $A_5$, denoted $\operatorname{Conj}(A_5)$, is the subgroup of $\operatorname{Aut}(A_5)$ consisting of automorphisms of the form $\gamma_s:=s(-)s^{-1}$ where $s\in A_5$. Explicitly, $\gamma_s(x)=sxs^{-1}$ for any $x\in A_5$.
\begin{enumerate}[label=\alph*)]
	\item Prove that the function $\gamma:A_5\to \operatorname{Conj}(A_5)$, taking $s\in A_5$ to $\gamma_s$, is a surjective homomorphism.
	\item Prove that $A_5$ is isomorphic to $\operatorname{Conj}(A_5)$.
\end{enumerate}

Suppose $H$ is a group of order 15. Prove there does not exist a nontrivial group homomorphism $\phi:D_5\to H$, where $D_5$ is the dihedral group with ten elements.

Let $S_7$ denote the symmetric group.
\begin{enumerate}[label=\alph*)]
	\item Give an example of two non-conjugate elements of $S_7$ that have the same order.
	\item If $g\in S_7$ has maximal order, what is the order of $g$?
	\item Does the element $g$ that you found in part (b) lie in $A_7$? Fully justify your answer.
	\item Determine whether the set $\{h\in S_7\mid |h|=|g|\}$ is a single conjugacy class in $S_7$, where $g$ is the element you found in part (b).
\end{enumerate}

Let $G$ be the additive group ${\bf Z}_{2020}$ and let $H\subseteq G$ be the subset consisting of those elements with order dividing 20.
\begin{enumerate}[label=\alph*)]
	\item Prove $H$ is a subgroup of $G$.
	\item Find an explicit generator for $H$ and determine its order.
\end{enumerate}

Let $G$ denote the set of invertible $2\times 2$ matrices with values in a field. Prove $G$ is a group by defining a group law, identity element, and verifying the axioms. Credit is based on completeness.