I am having trouble with exercises from Chapter IV of Aluffi's Algebra: Chapter 0. After spending many hours on the following two, I guess I need some help:
Let $G$ be a finite group, and suppose there exists representatives $g_1,g_2,\dots,g_r$ of the $r$ distinct conjugacy classes in $G$, such that $g_ig_j=g_jg_i$. Prove that $G$ is commutative.
Since proving the commutativity of $G$ is the same as proving all conjugacy classes are singletons, I am guessing I need to apply the commutativity of $g_i$'s to show that all conjugacy classes are of the same size, since we already know there is always an element whose conjugacy class is a singleton. To do this, I guess we need to use the class formula for group actions, but I do not know how to apply it.
The second problem is:
$G$ is a finite group acting transitively on a set $S$. If $|S|>2$, then there is a $g\in G$ without fixed points in $S$.
The author gives a hint. Since $S$ is isomorphic to $G/H$ as $G$-sets for some subgroup $H$, we might as well let $S=G/H$. The hint says we should use a result we have proved: A finite group cannot be the union of the conjugacy classes of a proper subgroup. Again I do not know how to use this hint.
Thanks!
Best Answer
I will address the second problem. This follows directly from the fact that $G$ cannot be the union of the conjugates of any one of its subgroups. Then, using the fact that the isotropic subgroups are conjugate to each other (since it is transitive), we have that $G$ properly contains the union of all of the isotropic subgroups, i.e., there is an element $x\in G$ that is not in any isotropic subgroup, hence $x$ doesn't fix any element in $S$.