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\title{Math 720\\Fall 2010\\Homework 5}
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\centerline{\it Due Friday, October 29, 2010.}

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{\it This assignment will be devoted to showing that the only nonabelian simple group of order less than or equal to 100 is $A_5$.}

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\noindent 1. Suppose that $\vert G\vert=60$ and that $G$ is simple.
\begin{itemize}
\item[a)] (5 pt) Find the possibilities for the number of Sylow 2-subgroups ($n$) of $G$ and show that we only need concern ourselves with $n=5$ or $n=15$.
\item[b)] (5 pt) Show that if $n=5$ then $G$ is isomorphic to a subgroup of $S_5$ and conclude that $G\cong A_5$.
\item[c)] (5 pt) Show that if $n=15$ then there are two Sylow 2-subgroups (say $P$ and $Q$) that must intersect in a subgroup of $G$ of order 2.
\item[d)] (5 pt) If $H=N_G(P\bigcap Q)$ show that $4\vert\ \vert H\vert$ and conclude that $\vert H\vert\geq 12$. (Hint: any group of order 4 is abelian so $P\subseteq N_G(P\bigcap Q)$.)
\item[e)] (5 pt) Show that since the index of $H$ in $G$ less than or equal to 5, $G\cong A_5$.
\end{itemize}


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\noindent 2. Assume that $\vert G\vert\leq 100$ and that $G$ is simple and nonabelian. 
\begin{itemize}
\item[a)] (5 pt) Use the results from this (and previous assignments) to make a list of the possible orders of $G$.
\item[b)] (5 pt) Eliminate all possibilities except for $\vert G\vert=60$ or $90$ (most of these should be almost immediate).
\end{itemize}

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\noindent 3. Now assume that $\vert G\vert=90$ and is simple.
\begin{itemize}
\item[a)] (5 pt) Show that $G$ must have $6$ Sylow 5-subgroups.
\item[b)] (5 pt) Show that $G$ is necessarily isomorphic to a subgroup of $A_6$. (Hint: $G$ can be considered a simple subgroup of $S_6$; consider $G\bigcap A_6$).
\item[c)] (5 pt) Derive a contradiction by showing that $A_6$ has no subgroup of order $90$. (Hint: if $A_6$ has a subgroup of order $90$, look at the orbit of this group under conjugation action...what is the order of its normalizer?)
\end{itemize}

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\noindent 4. (5 pt) Conclude that if $\vert G\vert\leq 100$ and $G$ is nonabelian and simple, then $G\cong A_5$. 

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