proved by considering the elements of N which are fixed when G acts on N by conjugation. Since every central subgroup is normal, it follows that every minimal normal subgroup of a finite p-group is central and has order p. Indeed, the socle of a finite p-group is the subgroup of the center consisting of the central elements of order p. If G is a p-group, then so is G/Z, and so it too has a non-trivial center. The preimage in G of the center of G/Z is called the second center and these groups begin the upper central series. Generalizing the earlier comments about the socle, a finite p-group with order pn contains normal subgroups of order pi with 0 ≤ i ≤ n, and any normal subgroup of order pi is contained in the ith center Zi. If a normal subgroup is not contained in Zi, then its intersection with Zi+1 has size at least pi+1. Automorphisms The automorphism groups of p-groups are well studied. Just as every finite p-group has a non-trivial center so that the inner automorphism group is a proper quotient of the group, every finite p-group has a non-trivial outer automorphism group. Every automorphism of G induces an automorphism on G/Φ(G), where Φ(G) is the Frattini subgroup of G. The quotient G/Φ(G) is an elementary abelian group and its automorphism group is a general linear group, so very well understood. The map from the automorphism group of G into this general linear group has been studied by Burnside, who showed that the kernel of this map is a p-group. Examples p-groups of the same order are not necessarily isomorphic; for example, the cyclic group C4 and the Klein four-group V4 are both 2-groups of order 4, but they are not isomorphic. Nor need a p-group be abelian; the dihedral group Dih4 of order 8 is a non-abelian 2-group. However, every group of order p2 is abelian. The dihedral groups are both very similar to and very dissimilar from the quaternion groups and the semidihedral groups. Together the dihedral, semidihedral, and quaternion groups form the 2-groups of maximal class, that is those groups of order 2n+1 and nilpotency class n. Iterated wreath products The iterated wreath products of cyclic groups of order p are very important examples of p-groups. Denote the cyclic group of order p as W(1), and the wreath product of W(n) with