Conjugacy classes
G is a finite group with elements a and b. Let the conjugacy classes of these elements be A and B respectively and suppose |A|^2, |B|^2 < |G|. Prove that there is a non-identity element x in G s.t. x commutes with both a and b.
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Solution Summary
This is a proof regarding conjugacy classes and a non-identity element.
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Normal subgroup
Let N be a normal subgroup of G, a in G, and C the conjugacy class of a in G.
a) Prove that a in N if and only if C is contained in N.
b) If C_i is any conjugacy class in G, prove that C_i is contained in N or the intersection of C_i and N is empty.
c) Use the class equation to show that |N| = |C_1| + ...+ |C_k|, where C_1, ... , C_k are all the conjugacy classes of G that are contained in N