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\title{Algebra Homework Set 2}
\author{James Cummings (jcumming@andrew.cmu.edu)}


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   Due by class time on Wednesday 14 September. Homework must be typeset and submitted by email as a PDF file. 

\begin{enumerate}

\item Let $H$ and $N$ be groups and let $\psi$ be a HM from $H$ to $Aut(N)$. Define a binary operation
  on the set of pairs $(h, n)$ with $h \in H$ and $n \in N$ by
  \[
  (h_1, n_1) (h_2, n_2) = (h_1 h_2, \psi(h_2^{-1})(n_1) n_2)
  \]

  \begin{enumerate}

  \item  Prove that this operation makes the set of pairs into  a group $G$. 

  \item  Prove that if $H' = H \times 1$ and $N' = 1 \times N$ then $H' \le G$, $N' \lhd G$,
     $H' \simeq H$ and $N' \simeq N$.  

  \item  Prove that $G = H' N'$ and $H' \cap N' = 1$.   

  \end{enumerate} 
    
\item   Suppose that (as in a question from last week) we have $H \le G$, $N \lhd G$, $G = HN$ and
  $H \cap N = 1$. We saw that every element of $G$ can be written as $h n$ for unique $h \in H$ and
  $n \in N$. Prove that $G$ is isomorphic to the group constructed in Q1 for a suitable choice of HM $\psi$.


\item  Let $G = S_5$. For each relevant prime $p$:

  \begin{enumerate}

  \item   Describe and count the elements of $G$ whose order is a power of $p$.
    
  \item   Describe the Sylow $p$-subgroups of $G$, and determine how many there are.

  \item   Describe the normalisers of the Sylow $p$-subgroups of $G$ (it is enough to compute the
  normaliser of one Sylow $p$-subgroup, the others will be its conjugates)

  \end{enumerate}
  
\item Let $G$ be a group of order 20. Prove that:
  \begin{enumerate}
  \item $G$ has a normal subgroup $N$ of order $5$.
  \item $G$ has a subgroup $H$ of order $4$.
  \item $G = H N$ and $H \cap N = 1$.
  \end{enumerate}

  Use the information from the previous questions to classify groups of order 20 up to isomorphism,
  then check your answer against a table of groups of small order. 

\item Let $G$ be a group with distinct elements $a$ and $b$ such that $\vert a \vert = 2$,
  $\vert b \vert = n$ and $b^a = b^{-1}$. Prove that $a \notin \langle b \rangle$,
  and that the subgroup generated by $a$ and $b$ is dihedral of order $2 n$.
  
  
\item  Let $G$ be a nonabelian group of order $8$. Prove that $G$ has at least one element of order
  $4$ and no element of order $8$.
   We saw in HW1 that $Z(G) \simeq C_2$  and $G/Z(G) \simeq C_2 \times C_2$. Let $Z(G) = \langle a \rangle$. 
  Prove that for any element $b$ of order $4$, $b^2 = a$ and $\langle b \rangle \lhd G$. 

  Now fix an element $b$ of order $4$, and an element $c \notin \langle b \rangle$. Prove that
  $G = \langle b, c \rangle$,  $b^c = b^{-1}$. Prove that if $c$ has order $2$ then
  $G$ is dihedral of order $8$. 

  Now suppose that $c$ has order $4$. Prove that $G$ is not dihedral,
  $a$ is the only element of order $2$, $b c$ has order $4$,
  $c b = a b c$, and $G = \{ e, b, c, bc, a, ab, ac, abc \}$.
  Consult a table of groups of small order to see if there is a non-dihedral
  non-abelian group of order $8$
  (you have proved enough to show that there is at most one such group). 

  

\end{enumerate}

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