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\begin{document}

\def\ctln{\centerline}
\def\msk{\medskip}
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\def\ssk{\smallskip}
\def\hsk{\hskip.3in}
\def\ra{\rightarrow}
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\def\mt{{\mathcal T}}
\def\mb{{\mathcal B}}
\def\ms{{\mathcal S}}
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\def\bbr{{\mathbb R}}
\def\bbz{{\mathbb Z}}
\def\spc{$~$\hskip.15in$~$}

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\def\del{\partial}
\def\lra{$\Leftrightarrow$}
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\ctln{\bf Math 417 Problem Set 8}

\msk

Starred (*) problems are due {\bf either} Friday, October 19, if you want to get your graded 
worked back before the midterm, {\bf or} Monday, October 22, if you don't care if you do or not.


\begin{description}

\item{(*) 51.} (Gallian, p.209, \#59) If $\varphi:G\ra H$ is a homomorphism that is \ubr{onto},
show that if $g\in Z(G)$ then $\varphi(g)\in Z(H)$.

\msk

\item{51.} We showed that for $G=\bbz[x]$ = the integer polynomials under addition, and $a\in\bbz$, the
function $\varphi:\bbz[x]\ra\bbz$ given by $\varphi(p)=p(a)$ is a homomorphism. Describe the kernel 
of this homomorphism.

\msk

\item{52.} Suppose that $H$ and $K$ are (finite) subgroups of a group $G$ and that $|H|$ and $|K|$
are relatively prime. Show that $H \cap K = \{e_G\}$ .

\msk

\item{53.} Let $H$ be a subgroup of $G$ and let $a,b\in G$. Show that $aH = bH$ if
and only if $Ha^{-1} = Hb^{-1}$ .

\msk

\item{(*) 54.} (Gallian p.151, \#10) Let $a$ and $b$ be elements of a group $G$, and let $H$ and $K$ be subgroups of
$G$. If $aH = bK$, show that $H = K$ .

\ssk

\item{\spc} [One approach: show, first, that $aH=bH$ !]

\msk

\item{(*) 55.} (Gallian p.152, \#33) Let $H$ and $K$ be subgroups of a finite group $G$ with $K \sset H \sset G$ .
Prove that $[G:K] = [G:H]\cdot[H:K]$ .

\msk

\item{56.} (Gallian p.153, \#47) Show that in a finite group $G$ with $|G|$ odd, for every $a\in G$ the equation $x^2=a$ has exactly
one solution.

\ssk

\item{\spc} [Hint: show that the function (not a homomorphism!) $f:G\ra G$ given by $f(x)=x^2$ is \ubr{onto} !]

\msk

\item{57.} (Gallian p.151, \#20) (Without directly computing $\varphi(n)$,) Show that if $n>2$ then 
the group $\bbz_n^*$ of units mod $n$ (under multiplication) has even order.

\ssk

\item{\spc} [Hint: what is the order of $n-1$ ?]


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\end{document}


