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{\bf \Large 250A Homework 7}\\[7mm]
{\small Due Monday November 22}\\[4mm]
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\vspace{.4cm}
Let $p$ be prime throughout.
\vspace{.4cm}

\noindent
{\bf Question 1} 
Let $H\unlhd G$ such that $p\; \slashed| \; [G:H]$.
Show that every Sylow-$p$-subgroup of $G$ is contained in $H$. 

\vspace{.4cm}\noindent
{\bf Question 2} 
Let $K\unlhd G$ be a $p$-subgroup. Show $K$ is
contained in every Sylow-$p$-subgroup of $G$. 


\vspace{.4cm}\noindent
{\bf Question 3}
Decide upon the validity of the following statements. Give reasons
({\it i.e.} proofs or counterexamples) for your answers.
\begin{enumerate}
\item[(i)] If $H\leq G$ and $S$ is a Sylow-$p$-subgroup of $G$, 
then $S\cap H$ is a Sylow-$p$-subgroup of $H$.
\item[(ii)] If $S$ and $S'$ are Sylow-$p$-subgroups of $G$
and $G'$, respectively, then $S\times S'$ is a Sylow-$p$-subgroup
of $G\times G'$.
\item[(iii)] If $G$ has a Sylow-$p$-subgroup of order $p^n$, then all 
subgroups of order $p^{n-1}$ are conjugate. 
\end{enumerate}

 
\vspace{.4cm}\noindent
{\bf Question 4} 
If $H\leq G$ and $P$, $P'$ are distinct Sylow-$p$-subgroups of $H$,
show $P$ and $P'$ cannot be subgroups of the same Sylow-$p$-subgroup of $G$.
 
\vspace{.4cm}\noindent
{\bf Question 5} 
Recall that a simple group $G$ has no normal subgroups save for itself
and $1_G$. How many simple groups are there of the following
orders: 18, 56, 70, 225 and 700?  Give reasons for your answer in each case.
 
\vspace{.4cm}\noindent
{\bf Question 6}
Suppose $p$ and $q$ are distinct primes. Show there are no
simple groups of order $p^2q$.
 

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