Let be a group, a normal subgroup, and a conjugacy class such that . Prove that is a union of distinct -conjugacy classes, each of equal size. Deduce that a conjugacy class in which consists of even permutations is either a single conjugacy class or a union of two -conjugacy classes of equal size.
Considering the induced action of on by conjugation, is a union of -conjugacy classes. Now let be distinct -conjugacy classes. Because is a -conjugacy class, we have for some . ( denotes conjugation.) Now . Thus and are conjugate as subsets of , and we have . That is, all -conjugacy classes in have the same cardinality. If we fix , then, for some integer , where is the number of distinct -conjugacy classes in .
Now note that and that . We then have the following.
Since by the Second Isomorphism Theorem, .
Thus is the number of -conjugacy classes contained in , and all these classes have the same cardinality.
Consider now . If is a conjugacy class and , then by the above argument is a union of distinct -conjugacy classes of the same cardinality. Recall that is a maximal subgroup of . Since , we have that is either or . Thus the number of -conjugacy classes in is either 1 or 2, and these classes have the same size.