Nilpotent Groups

Note, however, thatN G, G/N and N nilpotent ⇏ G nilpotent.In this way, nilpotent groups are different to soluble groups.Example 7.6 Finite p-groups are nilpotent.Proof: Let G be a finite p-group, say |G| = p n .Weproceedbyinductionon |G|. If|G| =1,thenγ 1 (G) =G = 1 so G is nilpotent.Now suppose |G| > 1. Apply Corollary 2.41: Z(G) ≠ 1. Consider thequotient group G/Z(G). This is a p-group of order smaller than G, sobyinduction it is nilpotent, sayγ c+1 (G/Z(G)) = 1.Let π : G → G/Z(G)bethenaturalhomomorphism. ThenbyLemma7.4(ii),so γ c+1 (G) ker π =Z(G). Thusγ c+1 (G)π = γ c+1 (G/Z(G)) = 1,γ c+2 (G) =[γ c+1 (G),G] [Z(G),G]=1,so G is nilpotent.□The example illustrates that the centre has a significant role inthestudyof nilpotent groups. We make two further definitions:Definition 7.7 The upper central series of G, denoted(Z i (G)) for i 0, isthe chain of subgroups defined byZ 0 (G) =1;Z i+1 (G)/Z i (G) =Z(G/Z i (G)) for i 0.Suppose that Z i (G) G. Then Z(G/Z i (G)) is a normal subgroupof G/Z i (G), so corresponds to a normal subgroup Z i+1 (G) ofG containingZ i (G) bytheCorrespondenceTheorem. Inthiswaywedefineachainof subgroups1 =Z 0 (G) Z 1 (G) Z 2 (G) ··· ,each of which is normal in G. HereZ 1 (G) =Z(G).Definition 7.8 A central series for a group G is a chain of subgroupsG = G 0 G 1 ··· G n = 1such that G i is a normal subgroup of G and G i−1 /G i Z(G/G i )foralli.85