\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 $p$ be prime and let  $G$ be a  non-abelian group of   order $p^3$. Show that $Z(G)=[G, G]$
  and this  is a subgroup of order $p$. 


\item  Prove that every group of order $15$ is cyclic
 
\item  Prove there is no simple group of order $pq$ for distinct primes $p, q$.

\item  Repeat the previous exercise  for groups of  order $p^2 q$. 

\item  Let G be a  simple group of order $60$. Determine the numbers
   of Sylow $p$-subgroups for $p =  2, 3, 5$  (if you can't determine the
   exact values give as much information as you can). 

\item  Let $F$ be a finite field with $q$ elements and let $GL_n(F)$
     be the group of invertible linear maps from $F^n$ to $F^n$. What is
     the order of $G$?

\item   Prove or give a counterxample to the statement ``if $G$ is nilpotent,
        every subgroup of $G$ is also nilpotent''. 


\item  Prove that there are (up to isomorphism) only two non-abelian groups of order
     $8$ and describe them.
 
\item   Recall that if $G$ is a group then $G^{(n)}$ is defined by the
  induction  $G^{(0)} = G$, $G^{(n+1)} = [G^{(n)}, G^{(n)}]$.
  Compute this series of groups when $G = S_3$, $S_4, S_5$.






\end{enumerate}



\end{document}