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\title{Group Theory: Math30038, Sheet 4}
\date{GCS}
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\begin{enumerate}
\item Suppose that $G$ acts on a set $\Omega.$ If
$\alpha \in \Omega$, we let
\[ G_\alpha = \{ g \in G \mid \alpha g = \alpha\}.\]
Now suppose that $\beta, \gamma \in \Omega$ are such that
$\beta h = \gamma$ for some $h \in G$. Show that
$G_\gamma = h^{-1} G_\beta h.$
\item Let $P$ be a group of order $p^n$ where $p$
is a prime number. Suppose that $P$ acts on a finite set $Q$
of size $q$ where $p$ does not divide $q$. Show that this action of $P$
has a fixed point (i.e. there is $\alpha \in Q$ such that
$\alpha g = \alpha \forall g \in P$).
\item In how many essentially different ways can one colour
the edges of a regular octahedron using $c$ colours (where
each edge is monochromatic, and two colourings are deemed the
same if one can moved to the other by a rigid motion -- and reflections
are not allowed).
\item Let $G$ be a group with subgroups $H$ and $K$, each
of finite index in $G$. Prove that $H \cap K$ has finite index in
$G$.
\item Let $G$ be a group and suppose that $H \leq G$ and
$|G : H | < \infty.$ By considering the groups $g^{-1}Hg$
as $g$ ranges over $G$ (or otherwise), prove that $G$ has
a normal subgroup $N$ with $|G : N | < \infty$ and $N \leq H \leq G$.
\item Let $G$ be a group and suppose that $x, y \in G$.
Prove that $o(xy) = o(yx).$
\end{enumerate}
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