1. List all elements of the group of symmetries of the octahedraon
by their action on the vertices.
2. Recall the definition of $\tilda_F $, when $F$ is a collection
of bijections on a set $X$. Also recall the definition of of $f\in
F$ acting on $X$-choose-$2$, the collection of 2-subsets of $X$.
For the $6$-gon and the group of symmetries $D_6 $, compute
$\tilda $ for the action on the set $X$ of vertices of the $6$-gon
and $X$-choose-$2$.
3. Let $F$ be a concrete group of bijections on a set $X$. For a
fixed $x\in X$, let $H(x)=\{ f\in F| (x)f=f \}$. Show that $H(x)$
is a subgroup of $F$. It is called the stabilizer of $x$. What
does the relation $L$ signify, for this subgroup, and how many
equivalence classes are there?
4. A group is called commutative (or abelian) if $ab=ba$ for all
elements $a,b$ in the group.
(i) Let $G$ be a group such that $a^2 =e$ for all $a\in G$. Show
that $G$ is abelian.
(ii) What about $a^3 =e$ for all $a\in G$?
5. Let $H\subseteq G$ be a subgroup, and let $g\in G$ be a fixed
element. Let $gHg^{-1}=\{ ghg^{-1} | h\in H\ }$. Show that
$gHg^{-1}$ is also a subgroup of $G$. Interpret this for the
example $H\subseteq S_n $ that we worked out in the class.
6. What are all the subgroups of $Z_n $? And of $Z$?