A nontrivial p-group has nontrivial center

Theorem. Let p be any prime. A finite group G has a subgroup of order p if and only if p divides the order of G. In that case, the number of p-element subgroups H of G is congruent to 1 mod p.

Proof: Let |G|=N. Certainly the condition that p|N is necessary since [G:H] must be an integer. Assume now that N is a multiple of p. We shall show that the number of elements of order p in G is congruent to p-1 mod p. In particular it is nonzero. Since each such element is contained in a unique p-element group H, which contains p-1 elements of order p, it will also follow that the number of such groups H is congruent to 1 mod p (and in particular is not zero).

Consider the solutions of x1x2...xp=1 in G. There are exactly Np-1 of them, one for each choice of x1,x2,...,xp-1. Note that Np-1 is a multiple of p. But if (x1x2...xp) is a solution, so are all of its cyclic permutations. Since p is prime, the cyclic permutations are either all distinct or all equal, the latter being the case if and only if the xi are all equal. Hence the number of x in G such that xp=1 is a multiple of p. But x solves this equation if and only if it is either the identity or an order-p element. Therefore the number of order-p elements is congruent to -1 mod p, as claimed.

(Such arguments can be extended to give one of the standard proofs of the Sylow theorems. For our present purpose, the exact residue mod p of the number of x's or H's doesn't really matter, only the fact that these counts are not multiples of p.)

Theorem. A nontrivial p-group has nontrivial center.
(A p-group is a group whose order is some power of a prime p. A nontrivial p-group is thus one whose order is pn for some n>0.)

Proof: Let G be a nontrivial p-group, and P the set of order-p elements of G. We have seen that P is nonempty, and indeed that |P| is congruent to -1 mod p. Now consider the action of G on P by conjugation. The stabilizer under this action of any x in P is the centralizer C(x) of x, which is the subgroup of G consisting of all elements that commute with x. The orbit of x then has size [G:C(x)]. But G is a p-group, so [G:C(x)] is a power of p. Hence [G:C(x)] is either 1 or a multiple of p. Since |P| is not a multiple of p, it follows that at least one of the orbits is a singleton. Then C(x)=G, which is to say that x commutes with every element of G. We have thus found a nontrivial element x of the center of G, QED.

Tony Varilly notes a simpler proof: The center of any group is the union of the 1-element conjugacy classes in the group. For a p-group, the size of every conjugacy class is a power of p. Thus a nontrivial p-group always has at least p-1 non-identity conjugacy classes (since {1} is always a singleton conjugacy class). Hence the center is nontrivial, as claimed.