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\noindent
\parbox{2.25in}{\bf Math 4720 \& 5210}
\hfill {\Large \bf Homework \#3} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Sept. $13^\mathrm{th}$, 2021}

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%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 An Inverse Problem] Let $G$ be a group and let $\varphi:G \to G$ be defined by $\varphi(x)=x^{-1}$ for all $x \in G$. Show that $\varphi$ is a homomorphism if and only if $G$ is Abelian. 

When $G$ is Abelian (so that $\varphi$ is a homomorphism), is $\varphi$ an isomorphism? Why or why not?

{\it Note:} Since its domain and codomain match, we could call $\varphi$ an endomorphism if it's a homomorphism. And we could call it automorphism if it's an isomorphism.

\item[\#2 Being Productive] Let $A$ and $B$ be groups. Recall that $A \times B = \{ (a,b) \;|\; a\in A \mbox{ and } b \in B\}$ becomes a group when we multiply coordinatewise: $(a,b)(c,d)=(ac,bd)$.
\begin{enumerate}[(a)]
\item Prove that $A \times B \cong B \times A$.
\item Let $\pi_A: A \times B \to A$ be defined by $\pi_A(a,b)=a$. Show that $\pi_A$ is an epimorphism (an onto homomorphism). What is its kernel? What does the first isomorphism theorem say?
\item Show that $A \times B$ is Abelian if and only if {\it both} $A$ and $B$ are Abelian.  Is this still true if we swap out the word ``Abelian'' with the word ``cyclic''?
\end{enumerate}

\item[\#3 Cayley Permutes] Let $G$ be a group. We write $S(G) = \{ f:G \to G \;|\; f \mbox{ is a bijection } \}$ for the permutations on $G$.
Let $L_g:G \to G$ be defined by $L_g(x)=gx$ (i.e., left multiplication by $g$) and $R_g:G \to G$ be defined by $R_g(x)=xg$ (i.e., right multiplication by $g$).
\begin{enumerate}[(a)]
\item Show that $L_a \circ R_c = R_c \circ L_a$ for all $a,c \in G$ (just plug in $b \in G$ and compute). In other words, left and right multiplication operators commute.
         In fact, ``left and right multiplication operators commute'' is equivalent to which axiom?
\item Show that $L_g, R_g \in S(G)$ for every $g \in G$.
\item  {\bf [Grad. Problem]} Prove that $\varphi:G \to S(G)$ defined by $\varphi(g) = L_g$ is a monomorphism (i.e., one-to-one homomorphism). 

{\it Note:} You showed that $L_g \in S(G)$ in the last part, so $S(G)$ makes sense as a codomain. 

{\it Observation:} You just proved Cayley's Theorem: $G \cong \varphi(G)$  is a subgroup of $S(G)$ (i.e., every group is isomorphic to a subgroup of permutations).

\item Consider $D_4 = \langle x,y \;|\; x^4=1,y^2=1,(xy)^2=1\rangle = \{ 1,x,x^2,x^3,y,xy,x^2y,x^3y\}$. 

Relabel elements as integers: $1 \mapsto 1$, $x \mapsto 2$, $x^2 \mapsto 3$, \dots, $x^3y \mapsto 8$. Under this relabeling $L_x$ becomes the permutation $(1234)(5678)$. 

As determined by Cayley's theorem, find a subgroup of $S_8$ that is isomorphic to $D_4$.

\item  {\bf [Grad. Problem]} Define $\varphi_g(x) = L_g \circ R_{g^{-1}}(x) = gxg^{-1}$ (i.e., $\varphi_g$ is conjugation by $g$). Show $\varphi_g$ is an automorphism of $G$.

{\it Note:} $\mathrm{Inn}(G) = \{ \varphi_g \;|\; g \in G\}$ is the set of {\it inner automophisms} and $\mathrm{Aut}(G)$ is the set of automorphisms. It is not hard to show that 
$\mathrm{Inn}(G)$ is a subgroup of $\mathrm{Aut}(G)$ which in turn is a subgroup of $S(G)$.

\item  {\bf [Grad. Problem]}  Let $\varphi:G \to \mathrm{Inn}(G)$ be defined by $\varphi(g) = \varphi_g$ (i.e., $g$ maps to the conjugation by $g$ map). Show that the kernel of $\varphi$ is the center of $G$. (What does the isomorphism theorem say?) Also, show that $\mathrm{Inn}(G)$ is a normal subgroup of $\mathrm{Aut}(G)$.

{\it Note:} Automorphisms which aren't inner are called ``outer automorphisms''. In fact, $\mathrm{Aut}(G)/\mathrm{Inn}(G)$ is called the {\it outer automorphism group} of $G$.
\end{enumerate}

\item[\#4 Permutin' Some More] Find a permutation which conjugates $\sigma = (16)(253)(4879)$ to $\tau=(149)(23)(5867)$. 

\item[\#5 Quotients] Write down all of the subgroups and quotients of $\mathbb{Z}_{12}$.

\item[\#6 Quotients Again] Recall the subgroup lattice of $D_4$ (as given in class). 
\begin{enumerate}[(a)]
\item List the normal subgroups of $D_4$. For the non-normal subgroups, show why they fail to be normal by giving left coset which does not match its right coset.
\item Write down all of the quotients of $D_4$.
\end{enumerate}

\end{description}

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