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\author{Math 25}

\title{Homework 8 --- due Nov 14th}

\begin{document}

\maketitle
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There are two sections: A and B.
Please return each part
to the proper CA.

\renewcommand{\thesection}{\Alph{section}}

\section{Problems graded by Benjamin}

\subsection{} In each of the following exercises, you
are given a set $G$ and a composition law $(x,y) \mapsto x \otimes
y$. Determine whether $(G,\otimes)$ is a group and if it is, find the
unit element, give a formula for the inverse of an element, and
determine whether $G$ is abelian.

\begin{enumerate}
\item $G=\RR$ and $x\otimes y = x+y+xy$.
\item $G = (\RR\setminus \{0\}) \times \RR$ and $(x,u) \otimes (y,v)  =
  (xy,xv+u)$. 
\item $G=\{(x,y) \in \RR^2,\ x^2+y^2=1\}$ and $(x_1,y_1) \otimes
  (x_2,y_2) = (x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2)$.
\end{enumerate}

\subsection{} Let $G$ be a finite group and let $a,b \in G$ be such
that $(ab)^m=e$ for some $m \geq 1$. Prove that $(ba)^m=e$.

\subsection{} Let $G$ be a group such that for every $a \in G$, we
have $a^2=e$. Prove that $G$ is abelian.

\subsection{} 
\begin{enumerate}
\item Find all groups $G$ which have no non-trivial automorphisms.
\item Find all groups $G$ which have no non-trivial subgroups.
\end{enumerate}

\subsection{} Find all continuous homomorphisms $f: (\RR,+) \ra (\RR,+)$.

\section{Problems graded by Inna}

\subsection{} Prove that if $G$ is an infinite group, then it has an
infinite number of subgroups.

\subsection{} Prove that if $H$ is a subgroup of index $2$ of a group
$G$, then $H$ is a normal subgroup of $G$.

\subsection{} The center $Z=Z(G)$ of a group $G$ is the set of elements
$z \in G$ such that $zg=gz$ for every $g \in G$.
\begin{enumerate}
\item Prove that $Z$ is a subgroup of $G$.
\item Prove that if $G/Z$ is cyclic, then $G$ is abelian (and so $G=Z$).
\item Prove that if $G$ is not abelian, then $[G:Z] \geq 4$.
\item Define $C(G)=\{(x,y) \in G^2,\ xy=yx\}$. Prove that if $G$ is 
  finite and not abelian, then \[ \# C(G) \leq \frac{5}{8} (\# G)^2. \]
\end{enumerate}

\subsection{} Let $S_4=\on{Bij} (\{ a,b,c,d\})$. Prove that $S_4$ has a
  unique subgroup of order $12$. Call it $A_4$. Prove that $A_4$ has
  no subgroup of order $6$.

\subsection{} Prove that if $H$ is a subgroup of a finite group $G$
(and $H \neq G$), then $G \neq \cup_{g \in G} (gHg^{-1})$. 
\end{document}

