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\hfill \textbf{April  12, 2012}

\vspace{0.5 cm} \centerline{ \textbf{\large{Homework 5\hspace{1 cm}
Math 6303}}}

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\begin{enumerate}
\item
Let $\mbf{G}$ be a finite group. Show that $[G:cent(\mbf{G})]$
cannot be prime. (Hint: For any $g\in \mbf{G}$ one has that
$card(cent(\mbf{G}))|card(cent(g))$, i.e.
$[\mbf{G}:cent(g)]|[\mbf{G}:cent(\mbf(G)]$. Then use: $\mbf{G}$ is
abelian iff $cent(g)=cent(G)$ for some $g$.)

\item
Let $p$ be a prime number and let $\mbf{G}$ be a group of order
$p^2$. Prove that $\mbf{G}$ is abelian.

\item
(This is the Generalized  Cayley's Theorem.) Let $\mbf{H}$ be a
subgroup of $\mbf{G}$ and let $X$ be the set of left cosets of
$\mbf{H}$ in $\mbf{G}$. Show that there exists a normal subgroup
$\mbf{N}$ of $\mbf{G}$ such that $\mbf{N}\subseteq \mbf{H}$ and
$\mbf{G}/\mbf{N}$ is isomorphic to a subgroup of $Sym(X)$. (Hint:
$\rho:\mbf{G}\to Sym(X), \rho(g)(h\mbf{H})=(gh)\mbf{H}$ is a
representation of $\mbf{G}$ where $\ker(\rho)\subseteq \mbf{H}$.)

\item
Let $\mbf{G}$ be a group, $\mbf{H}$ a subgroup of index $n$ in
$\mbf{G}$. Prove that there is a normal subgroup $\mbf{N}$ of
$\mbf{G}$  such that $\mbf{N}\subseteq \mbf{H}$ and
$[\mbf{G}:\mbf{N}]$ divides $n!$.

\item
(This exercise generalizes that subgroups of index $2$ are
normal.) Suppose that $\mbf{G}$ is a finite group and $p$ is the
smallest prime that divides $card(\mbf{G})$. Let $\mbf{H}$ be a
subgroup of $\mbf{G}$ such that $[\mbf{G}:\mbf{H}]=p$. Prove
that $\mbf{H}$ is normal in $\mbf{G}$.
\item Hungerford, p. 96, Exercises 1, 2, 3, 6
\end{enumerate}







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