Section 1.7. Group Actions 69 Proof. We prove the theorem by induction on m ≥ 1, where |G| = pm. The base step m = 1 is true, for Lagrange’s Theorem shows that every nonidentity element in a group of order p has order p. Let us now prove the inductive step. If x ∈ G, then the number of conjugates of x is |xG| = [G : CG(x)], where CG(x) is the centralizer of x in G. If x / ∈ Z(G), then xG has more than one element, and so |CG(x)| |G|. We are done if p | |CG(x)|, for the inductive hypothesis gives an element of order p in CG(x) ⊆ G. Therefore, we may assume that p |CG(x)| for all noncentral x ∈ G. Better, since p is prime and |G| = [G : CG(x)]|CG(x)|, Euclid’s Lemma gives p | [G : CG(x)]. After recalling that Z(G) consists of all those elements x ∈ G with |xG| = 1, we may use Proposition 1.106 to see that |G| = |Z(G)| + i [G : CG(xi)], where one xi is selected from each conjugacy class having more than one element. Since |G| and all [G : CG(xi)] are divisible by p, it follows that |Z(G)| is divisible by p. But Z(G) is abelian, and so Proposition 1.84 says that Z(G), and hence G, contains an element of order p. • Definition. The class equation of a finite group G is |G| = |Z(G)| + i [G : CG(xi)], where one xi is selected from each conjugacy class having more than one element. Definition. If p is prime, then a group G is called a p-group if every element has order a power of p. Proposition 1.112. A finite group G is a p-group if and only if |G| = pn for some n ≥ 0. Proof. Let G be a finite p-group. If |G| = pn, then there is some prime q = p with q | |G|. By Cauchy’s Theorem, G contains an element of order q, a contradiction. The converse follows from Lagrange’s Theorem. • We have seen examples of groups whose center is trivial for example, Z(S3) = {1}. For finite p-groups, however, this is never true. Theorem 1.113. If p is prime and G = {1} is a finite p-group, then the center of G is nontrivial Z(G) = {1}. Proof. Consider the class equation |G| = |Z(G)| + i [G : CG(xi)]. Each CG(xi) is a proper subgroup of G, for xi / ∈ Z(G). Since G is a p-group, [G : CG(xi)] is a divisor of |G|, hence is itself a power of p. Thus, p divides each

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