1. Let $H\subset G$ be a subgroup of the group $G$. Let $g\in G$
be an arbitrary element of $G$. Define
\[ gHg^{-1} =\{ ghg^{-1} | h\in H \} \]
Show that $gHg^{-1} $ is a subgroup of the same cardinality as
$H$.
2. A subgroup $H$ is called normal if $gHg^{-1} =H$ for all $g\in
G$. Let $H$ be arbitrary, and $S=\{ aH|a\in G\} $ be the
collection of left cosets of $H$ in $G$. Consider the following
``multiplication'' on $S$: given $aH$ and $bH$, define $aH.bH$ as
$abH$. Show that this multiplication is well defined if and only
if $H$ is normal. Hint: Here well-defined-ness means that the
product must be independent of the coset representatives chosen.
In other words, if $aH=a'H$ and $bH=b'H$ then we should expect
$abH=a'b'H$.
3. Let $G$ act on the set $S$ and $s\in S$. Let the stabilizer of
$s$ be the subgroup $H(s)$. Let $t=s.g$, then show that the
stabilizer of $t$, $H(t)=g^{-1} H(s) g$.
4. Construct an injective homomorphism of the group $Bij(S)\times
Bij (T)$ into the larger $Bij(S\times T)$. In other words, given
$f\in Bij(S)$ and g\in Bij(T)$ give an interpretation for the
tuple $(f,g)$ as a bijection on $S\times T$ and show that this
interpretation preserves the group operations.
5. Let $S$ be the set of vertices of a 5-gon with $D_5 $ acting on
it. Compute the number of orbits for S-choose-2. Compute the same
for S-choose-2 and S-choose-3 on the 6-gon.
6. Let $G$ act on $S$. Let $H$ be the collection of all elements
$g$ of $G$ such that $s.g=s$ for all $s\in S$. Show that $H$ is a
normal subgroup of $G$. We say that the action of $G$ is
``faithful'' if $H= \{ e\}$.