I am having trouble with exercises from Chapter IV of [Aluffi's *Algebra: Chapter 0*](http://www.mimuw.edu.pl/~jarekw/pdf/Algebra0TextboookAluffi.pdf). After spending many hours on the following two, I guess I need some help:

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> 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.
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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:

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> $G$ is a finite group acting transitively on a set $S$. If $|S|>2$, then there is a $g\in G$ without fixed points in $S$.
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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!