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\title{Algebra Homework Set 1}
\author{James Cummings (jcumming@andrew.cmu.edu)}


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   Due by class time on Wednesday 7 September. Homework must be typeset and submitted by email as a PDF file. 

\begin{enumerate}

\item   Prove that if $H \le G$ and $N \lhd G$ then $H N = N H$ and $H N \le G$. Prove further that
       if $H \cap N = 1$ and $G = H N$, then every element of $G$ can be written uniquely as
       $h n$ for some $h \in H$ and $n \in N$. Prove that if additionally $H \lhd G$, then
       $G \simeq H \times N$. 
       
\item  As we noted in class, any group $G$ acts on the set of its subgroups by conjugation, explicitly
       $g \cdot H = H^g = g H g^{-1}$. The stabiliser of $H$ in this action is called the {\em normaliser}
       and is written $N_G(H)$. Prove that if $H$ and $K$ are subgroups and $K \le N_G(H)$ then
       $H K = K H \le G$

\item  Let $G$ act on $X$ and let $x$ and $y$ lie in the same orbit. Prove that the stabilisers of $x$ and $y$ are conjugate subgroups.

\item  Let $G$ be a group, let $H \le G$ and let $X$ be the set of left cosets of $H$ in $G$. Let $G$ act on $X$ by setting
       $g \cdot C = g C$. Describe:

\begin{itemize}

\item  The stabiliser of $H$.

\item  The kernel of the action (when viewed as a HM from $G$ to the permutation group $\Sigma_C$ of $C$),
       that is to say the set of $g$ such that $g C = C$ for every left coset $C$. 

\end{itemize} 


\item  A group $G$ is {\em simple} if $G \neq 1$ and the only normal subgroups of $G$ are $1$ and $G$. 
       Prove that if $G$ is simple,  $H < G$ (that is $H \le G$ and $H \neq G$) and $C$ is the set of left cosets of
       $H$ then $G$ is isomorphic to some subgroup of $\Sigma_C$. 

\item  Let $F$ be a finite field with $q$ elements.\footnote{As we see later in the course, $q$ must be a prime power 
       and up to isomorphism there is one field of size $q$ for every prime power $q$} Find the order of the group of
       invertible $n \times n$ matrices with entries in $F$.

\item  Let $p$ be prime and let $G$ be the set of complex numbers $z$ such that $z^{p^n} = 1$ for some $n > 0$.
       Prove that $G$ forms a group under multiplication. Determine the set of subgroups of $G$.  
 
\item  Prove that if $G/Z(G)$ is cyclic then $G$ is abelian. Give an example to show that $G/Z(G)$ being abelian
       does not imply that $G$ is abelian. Prove that if $G$ is a non-abelian group of order $p^3$ for some
       prime $p$ then $Z(G) \simeq C_p$ and $G/Z(G) \simeq C_p \times C_p$.   




\end{enumerate}

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