Representatives for conjugacy classes

Posted: February 19, 2010 in Elementary Algebra; Problems & Solutions, Groups and Fields
Tags: , ,

Problem. Let G be a finite group and g_1, \cdots, g_r a system of representatives  for the conjugacy classes of G. Prove that if the g_i pairwise commute, then G is abelian.

Solution. We will denote by C(g) the centralizer of g \in G in G. Suppose that G is not abelian. Then |G| > r and we may assume that Z(G) = \{g_1, \cdots, g_s \}, for some 1 \leq s < r. Since the g_i pairwise commute, we have g_i \in C(g_j), for all i and j, and hence |C(g_j)| \geq r for all j. Thus \displaystyle [G: C(g_j)] \leq \frac {|G|}{r}, for all j. So by the class equation \displaystyle |G| = s + \sum_{j = s + 1}^r[G: C(g_j)] \leq s + \frac {r - s}{r}|G|, which gives us the contradiction |G| \leq r. \ \Box

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