\documentclass{amsart}

\title{Algebra Homework Set II}


\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. 

    
    NOTE: This is one is shorter because it's late. Apologies for the delay.

    $C_n$ denotes a cyclic group of order $n$.


\begin{enumerate}


\item  Use Sylow's theorem to show that there is no simple group of 
       any of the following orders: $20$, $26$, $34$, $48$.


\item  Describe the automorphism group $Aut(C_5)$.
       Find all the HMs from $G$ to $Aut(C_5)$ for
\begin{enumerate}
\item $G = C_4$.
\item $G = C_2 \times C_2$.
\end{enumerate} 
        

\item  Suppose that $G = H N$ where $H \le G$, $N \lhd G$ and $H \cap N = \{ e \}$.
       
\begin{enumerate}
\item  Show that every element of $G$ can be written as $h n$ for  unique elements
     $h \in H$ and  $n \in N$.
\item  Express $h_1 n_1 h_2 n_2$ where $h_i \in H$, $n_i \in N$ in the form $h n$ with
    $h \in H$, $n \in N$.
\end{enumerate}




\item  Suppose that $H, N$ are groups and $\psi: H \rightarrow Aut(N)$ is a HM.
   Define a binary operation $*$ on the set of ordered pairs $H \times N$ as follows:
\[
   (h_1, n_1) * (h_2, n_2) = (h_1 \times_H h_2, \psi(h_2^{-1})(n_1) \times_N n_2).
\]
  Prove that this  binary operation makes $H \times N$ into a group. 

\item  Explain the relationship between the previous two questions. Use Sylow's theorem
    and the ideas of the previous two questions to classify groups of order $20$
    up to isomorphism.

    Hint: there may be unexpected IMs among the groups that you construct, be careful.





\end{enumerate}



\end{document}
