
\documentclass[12pt]{article}
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\usepackage{amssymb}
\usepackage{dutchcal}

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\begin{document}

\def\ctln{\centerline}
\def\msk{\medskip}
\def\bsk{\bigskip}
\def\ssk{\smallskip}
\def\hsk{\hskip.3in}
\def\ra{\rightarrow}
\def\ubr{\underbar}

\def\mt{{\mathcal T}}
\def\mb{{\mathcal B}}
\def\ms{{\mathcal S}}
\def\mu{{\mathcal U}}
\def\mv{{\mathcal V}}

\def\bbr{{\mathbb R}}
\def\bbz{{\mathbb Z}}
\def\spc{$~$\hskip.15in$~$}

\def\sset{\subseteq}
\def\del{\partial}
\def\lra{$\Leftrightarrow$}
\def\bra{$\Rightarrow$}



%%\UseAMSsymbols

\ctln{\bf Math 417 Problem Set 3}

\msk

Starred (*) problems are due Friday, September 14.


\begin{description}


\item{14.} Give an \underbar{example} of a group $G$ and $a,b\in G$ so that 
$(ab)^4=a^4b^4$, but $ab\neq ba$. 

\ssk

\item{\spc} [Hint: Problem \#11 might help? Slightly bigger challenge: try the
same thing with the 4's replaced by 3's !]
 

\msk

\item{(*) 15.} (Gallian, p.58, \#51) Show that, if we had `weakened' the definition of a 
group $G$ to (1) there is an $e\in G$ with $ge=g$ for every $g\in G$, (2) inverses
exist, and (3) the group operation is associative, then we can \underbar{prove} that
$eg=g$ for every $g\in G$ (i.e, the other half of the definition of an identity
automatically holds).

\msk

\item{16.} (Gallian, p.57, \#43, sort of) Show that if $R$ is a rotation of $\bbr^2$
around the origin, and $M$ is a reflection over a line through the origin, then for 
any $k\in\bbz$ we have $R^kMR^k=M$.

\ssk

\item{\spc} [Hint: Problem \#1 might help!]

\msk

\item{(*) 17.} (Gallian, p.57, \#39) If $G$ is a group, and for every $a,b,c,d,x\in G$
we have $axb=cxd$ implies that $ab=cd$, show that then for every $u,v\in G$ we 
have $uv=vu$. ('A middle cancellation law implies commutativity.')

\ssk

\item{\spc} [Hint: Find an $x$ so that $uxv=vxu$ !]

\msk

\item{18.} (Gallian, p.69, \# 4) Show that if $G$ is a group and $a\in G$, 
then $|a|=|a^{-1}|$. 

\msk

\item{(*) 19.} If $G$ is a group and $a\in G$, and if $|a|<\infty$ and $\textrm{gcd}(k,|a|)=1$, 
show that then $|a^k|=|a|$.

\msk

\item{20.} (Gallian, p.74, \#77) Show that if $G$ is a group with $g\in G$ and $n=|g|<\infty$,
and $k$ is relatively prime to $n$, then there is an $h\in G$
with $g=h^k$. 

\ssk

[Hint: This should be true even if we replace $G$ with a 
\ubr{subgroup} of $G$ which contains $g$, e.g., $H=\langle g\rangle$ !]


\end{description}
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\end{document}


