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\title{Math 420-620\\Fall 2012\\Homework 10}
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\centerline{\it Due Wednesday November 7, 2012.}

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\noindent 1. (5 pt) Consider the finite abelian group

\[
\mathbb{Z}_{108}\oplus\mathbb{Z}_{24}\oplus\mathbb{Z}_{1125}\oplus\mathbb{Z}_{420}\oplus\mathbb{Z}_{620}.
\]

\begin{itemize}
\item[a)] (5 pt) Find the invariant factor decomposition for this group.
\item[b)] (5 pt) Find the elementary divisor decomposition for this group.
\end{itemize}

\centerline{}

\noindent 2. (5 pt) Let $K$ be the group of order 2 and $H$ be abelian. Suppose that $\phi:K\longrightarrow\text{Aut}(H)$ takes the nonidentity element to the automorphism of $H$ that takes each element to its inverse.
\begin{itemize}
\item[a)] (5 pt) Find necessary and sufficient conditions on $H$ so that $H\rtimes_{\phi} K\cong H\times K$.
\item[b)] (5 pt) What can you say about $H\rtimes_{\phi} K$ in the case where $H$ is cyclic?
\item[c)] (5 pt) Find all groups of order 8 that cannot be written as the semidirect product of two of its proper subgroups. 
\end{itemize}  

\centerline{}

\noindent 3. Show that $\text{S}_n$ is not solvable if $n\geq 5$ (you may use the fact that $\text{A}_n$ is simple if $n\geq 5$).

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