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\title{Algebra Homework Set 3}
\author{James Cummings (jcumming@andrew.cmu.edu)}


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   Due by class time on Wednesday 28 September. Homework must be typeset and submitted by email as a PDF file. 

\begin{enumerate}

\item Prove that a finite group $G$ is solvable if and only if its composition factors are cyclic groups of
     prime order. 

\item  Compute the derived series of $S_4$. Hint: Keep in mind that $[G, G]$ is the smallest normal
  subgroup of $G$ with an abelian quotient, this can save you computing too many commutators.
    Is the derived series of $S_4$ a composition series? Find the composition factors of $S_4$. 
    Is $S_4$ nilpotent?

\item  Prove that if $G$ is a non-trivial finite $p$-group then $[G, G] < G$.
    Hint: Use $Z(G) \neq 1$ to power an induction. 

    
\item  Suppose that $G$ is a simple group of order $60$. Derive as much information about the
    Sylow $p$-subgroups of $G$ as you can: their number, their structure, their normalisers.
    Hint:  You can use HW1Q5 to get some information.

\item  Prove that if $p$ and $q$ are distinct primes there is no simple group of order $p^2 q$.
    
\item  Let $S$ be the subgroup of $\Sigma_{\mathbb N}$ generated by the set of transpositions. Prove that
  $S \neq \Sigma_{\mathbb N}$. Let $A$ be the subgroup of $\Sigma_{\mathbb N}$ generated by the set of products
  of two transpositions, prove that $A = [S, S]$,  $[S : A] = 2$ and $A$ is simple. Hint: It is a standard fact that
  $A_n$ is simple for $n \ge 5$, this statement and/or its proof may be helpful.  
  
\item Let $G$ be the group of symmetries of the Euclidean plane. Prove that $G$ is solvable.
  Hint: You may find it helpful to note that if $S$ and $T$ are symmetries then $S T S^{-1}$
  is the map which moves $S(P)$ to $S(T(P))$ for each point $P$, so in some sense it's just a shifted
  version of $T$. Optional not for credit brainteaser: Is the symmetry group of Euclidean
  $3$-space solvable? 
  
\end{enumerate}

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