\documentclass{amsart}

\title{Algebra Homework Set I}


\author{James Cummings}



\begin{document}


\maketitle

    You may collaborate on this homework set, but must write up your solutions
    by yourself. Please contact me by email if you are puzzled by something,
    would like a hint or believe that you have found a typo. 

    





\begin{enumerate}



\item   Let $G$ be a group and let $H, K \le G$.  The {\em double cosets of
    $H$ and $K$} are
    the subsets of $G$ of the form $H a K$ where $a \in G$. Prove that the 
    double cosets form a partition of $G$. Is it true in general that all the
    double cosets have the same cardinality? (Either prove it or give a
    counterexample)

\item   Let $G$ be a group and let $H, K \le G$. Prove that if
    $h_1, h_2 \in H$ then $h_1 K = h_2 K \iff h_1(H \cap K) = h_2 (H \cap K)$.
    Use this  to prove that if $H, K$ are both finite then the set of products
    $H K$ has cardinality $\vert H \vert \times \vert K \vert / \vert H \cap K
    \vert$.

\item  Recall from class that for any group $G$ 

\begin{enumerate}

\item  An {\em automorphism (AM)} of $G$ is an isomorphism from $G$ to $G$.

\item  For any $g \in G$, the map $h \mapsto h^g$ is an AM of $G$, where as
  usual $h^g = g h g^{-1}$. 

\item  $N \lhd G$ iff $N^g = N$ for all $g \in G$.

\end{enumerate}

   A subgroup $H$ of $G$ is said to be {\em characteristic} iff $\alpha(H) =
   H$ for every AM $H$ of $G$ (so that clearly any characteristic ssubgroup is
   normal). In this case we write ``$H\,char\,G$''. 

\begin{enumerate}

\item  Prove that $Z(G)$ is a characteristic subgroup of $G$. 

\item  Give an example of a group $G$ and a normal subgroup of $G$ which is 
    not characteristic. Hint: every subgroup of an abelian group is normal. 

\item  Let $G$ be a group. Prove that if $A\,char\,B$ and $B\,char\,G$ then
    $A\,char\,G$, and that if $A\,char\,B$ and $B \lhd G$ then $A \lhd G$. 

\item  Give an example of a group $G$ and subgroups $B \lhd G$ and $A \lhd B$
  such that $A$ is not normal in $G$.

\end{enumerate}


\item Let $G$ be a group and let $H \le G$ with $[G : H]= 2$. 

\begin{enumerate}

\item  Prove that $H \lhd G$, and that if $h \in H$ then the $G$-conjugacy
  class of $g$ is a subset of $H$.

\item  Prove that if $h \in H$ then {\em either} the $G$-conjugacy class of
  $h$ equals the $H$-conjugacy class of $h$ {\em or} the $G$-conjugacy class
  of $h$ is the union of two disjoint $H$-conjugacy classes.

\end{enumerate}


\item   Show that if $G$ is a group such that $a^2 = e$ for all $a \in G$,
  then $G$ is abelian.  (Trickier) Produce an example of a non-abelian
  group $G$ such that $a^3  = e$ for all $a \in G$.


\item   Prove that if $G/Z(G)$ is cyclic then $G$ is abelian (so in fact
        $Z(G) = G$). Give an example of a non-abelian group such that
        $G/Z(G)$ is abelian.


\item   Find $Z(G)$ when $G$ is the group of non-singular $n \times n$ real
  matrices under matrix multiplication (and $n \ge 2$).

   







\end{enumerate}



\end{document}