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\ctln{\bf Math 417 Problem Set 11}

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Starred (*) problems are due Friday, November 30.


\begin{description}

\item{74.} If $H,K\subseteq G$ are subgroups of $G$, then we can define the \ubr{product} (sets)

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\ctln{$HK = \{hk\ :\ h\in H,\ k\in K\}$ \hskip.2in and \hskip.2in $KH = \{kh\ :\ k\in K,\ h\in H\}$ .}

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\item{\spc} Show that $HK$ is a subgroup of $G$ \lra\ $HK=KH$.

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\item{(*) 75.} Show that if $H,K\subseteq G$ are subgroups of $G$, and $HK$ is also 
a subgroup, then $|H|\cdot|K|=|HK|\cdot|H\cap K|$. 

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[Hint: show that if you pick coset representatives $A=\{a_1(H\cap K),\ldots,a_n(H\cap K)\}$ of the subgroup
$H\cap K$ in $H$, then the map $A\times K\rightarrow HK$ given by $(a(H\cap K),k)\mapsto ak$ is 
a \ubr{bijection}.]

\msk

\item{76.} (Gallian, p.399, \# 11) Show that if $G$ is a group and $G$ is finite and odd, then 
for every $g\in G$ with $g\neq e_G$ we have that $g^{-1}$ is not conjugate to $g$, i.e., 
$g^{-1}$ is not in $\textrm{cl}(g)$.

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\item{\spc} [Hint: $(xgx^{-1})^{-1}=x(g^{-1})x^{-1}$ would also always be conjugate to $g$...]

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\item{(*) 77.} If $|G|=p^n$ with $p$ prime, show that for every $k$, $1\leq k\leq n$,
there is a \ubr{normal} subgroup $N\leq G$ with $|N|=p^k$.

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\item{\spc} [Hint: take the quotient by some element of the center of $G$, and use induction!]

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\item{78.} Sylow's first theorem implies that the symmetric group $S_5$ (with $|S_5|=5!=2^3\cdot 3\cdot 5$)
has a subgroup of order $8$. Find one! Is the subgroup you found a normal subgroup of $S_5$?

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\item{\spc} [Hint: $S_4$ also has a subgroup of order 8. And we know a group of order 8 that acts on 
a certain set of 4 points...]

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\item{(*) 79.} In class we showed that for $p$ a prime, $|GL(2,\bbz_p)| = p(p-1)(p^2-1)$. 
So, for example, $|GL(2,\bbz_5)|=5\cdot 4\cdot 24 = 480$, and so (by Sylow) 
$GL(2,\bbz_5)$ must have elements
of order $3$ and of order $5$. Find some! Are the subgroups that they generate normal?

\msk

\item{80.} Show that every group of order $175$ is abelian. 

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\item{81.} According to Sylow theory, how many 5-, 7-, and 11-Sylow subgroups
\ubr{could} a group of order $5^2\cdot 7\cdot 11$ have?


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