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\def\ctln{\centerline}
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\def\ra{\rightarrow}
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\def\bbz{{\mathbb Z}}
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\ctln{\bf Math 417 Problem Set 9}

\msk

Starred (*) problems are due Friday, November 9.


\begin{description}

\item{(*) 58.} If $\varphi:G\ra H$ is a \ubr{surjective} homomorphism and $N\leq G$
is a \ubr{normal} subgroup of $G$, show that $\varphi(N)\leq H$ is a normal subgroup of $H$.
Show, on the other hand, that if $\varphi$ is not surjective, then $\varphi(N)$ 
need not be a normal subgroup (hint: $G$ is a normal subgroup of $G$ !).

\msk

\item{59.} If $H,K\leq G$ are \ubr{normal} subgroups of the group $G$, show that 
$H\cap K\subseteq G$ is also a normal subgroup of $G$, and there is an \ubr{injective}
homomorphism $G/(H\cap K)\ra (G/H)\oplus(G/K)$.

\ssk

\item{\spc} [Note: This is a generalization of our work in class on $\bbz_{21}$ and $\bbz_{21}^*$ .]

\msk

\item{(*) 60.} (Gallian, p.222, \# 42) Show that if $N,K\leq G$ are \ubr{normal}
subgroups of $G$ and $K\leq N$, then $N/K$ is a normal subgroup of 
$G/K$, and $(G/K)/(N/K) \cong G/N$. [This is the ``Third Isomorphism Theorem'' 
of Emmy Noether. One approach: start by looking at the `natural' map $G\ra G/N$.]

\msk

\item{61.} If $G$ is a group, show that $H=\{(g,g)\ :\ g\in G\}$ is a normal 
subgroup of $G\oplus G$ \lra\ $G$ is \ubr{abelian}; when $H$ is normal, 
show that $(G\oplus G)/H$ is isomorphic to $G$.

\ssk

\item{\spc} [Hint: how would you build a homomorphism $G\oplus G\ra G$ so that $H$ would be the kernel? Note
that at this point in the problem you can \ubr{assume} that $G$ is abelian!]

\msk

\item{62.} (Gallian, p.202, \# 35 (sort of)) Show that the functions
$\varphi,\psi:\bbz\oplus\bbz\oplus\bbz\ra\bbq$ given by $\varphi(a,b,c)=3^a6^b10^c$
and $\psi(a,b,c)=3^a6^b12^c$ are homomorphisms (where the domains are groups under 
addition and the codomains are groups under multiplication), and that 
$\varphi$ is injective, while $\psi$ is \ubr{not}.

\msk

\item{(*) 63.} (Gallian, p.202, \# 37) If $H$ is a normal subgroup in $G$ and $G$ is finite,
and $g\in G$, show that the order of $gH$ in $G/H$ divides the order of $g$ in $G$.

\msk

\item{64.} Show that in the symmetric group $S_n$, every \ubr{commutator} $\alpha\beta\alpha^{-1}\beta^{-1}$
is an element of the subgroup $A_n$ = the alternating group.
Show, in addition, that
every 3-cycle $(a,b,c)$ can be written as a 
commutator $\alpha\beta\alpha^{-1}\beta^{-1}$ . Conclude that every element of $A_n$
can be written as a \ubr{product} of commutators. 

\msk

\item{65.} Use problem \#64 to show that if $\varphi:S_n\ra G$ is a homomorphism from the 
symmetric group
to an \ubr{abelian} group $G$, then $\varphi(A_n)=\{e_G\}$ and so $\varphi(S_n)$ is either 
the trivial subgroup or isomorphic
to $\bbz_2$.


\end{description}
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