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

Lemma 7.9 LetG = G 0 G 1 ··· G n = 1be a central series for G. Thenforalli:γ i+1 (G) G i and Z i (G) G n−i .Proof: First observe that γ 1 (G) =G = G 0 . Suppose that γ i (G) G i−1for some i. Ifx ∈ γ i (G) andy ∈ G, thenG i x ∈ G i−1 /G i Z(G/G i ),so G i x commutes with G i y.Thereforeso [x, y] ∈ G i .HenceG i [x, y] =(G i x) −1 (G i y) −1 (G i x)(G i y)=G i ,γ i+1 (G) =[γ i (G),G] G i .Thus, by induction, the first inclusion holds.Now, Z 0 (G) =1 = G n . Suppose that Z i (G) G n−i . Since (G i )isacentral series for G,Thus if x ∈ G n−i−1 and y ∈ G, thenG n−i−1 /G n−i Z(G/G n−i ).G n−i x and G n−i y commute; i.e., [x, y] ∈ G n−i .Hence [x, y] ∈ Z i (G), so Z i (G)x and Z i (G)y commute. Since y is an arbitraryelement of G, wededucethatZ i (G)x ∈ Z(G/Z i (G)) = Z i+1 (G)/Z i (G)for all x ∈ G n−i−1 .ThusG n−i−1 Z i+1 (G) andthesecondinclusionholdsby induction.□We have now established the link between a general central series andthe behaviour of the lower and the upper central series.Theorem 7.10 The following conditions are equivalent for a group G:(i) γ c+1 (G) =1 for some c;(ii) Z c (G) =G for some c;(iii) G has a central series.Thus these are equivalent conditions for a group to be nilpotent.86

Proof: If G has a central series (G i )oflengthn, thenLemma7.9givesγ n+1 (G) G n = 1 and Z n (G) G 0 = G.Hence (iii) implies both (i) and (ii).If Z c (G) =G, thenG =Z c (G) Z c−1 (G) ··· Z 1 (G) Z 0 (G) =1is a central series for G (as Z i+1 (G)/Z i (G) =Z(G/Z i (G))). Thus (ii) implies(iii).If γ c+1 (G) =1, thenG = γ 1 (G) γ 2 (G) ··· γ c+1 (G) =1is a central series for G. (Forifx ∈ γ i−1 (G) andy ∈ G, then[x, y] ∈ γ i (G),so γ i (G)x and γ i (G)y commute for all such x and y; thusγ i−1 (G)/γ i (G) Z(G/γ i (G)).) Hence (i) implies (iii).□Further examination of this proof and Lemma 7.9 shows thatγ c+1 (G) =1 if and only if Z c (G) =G.Thus for a nilpotent group, the lower central series and the upper centralseries have the same length.Our next goal is to develop further equivalent conditions for finitegroupsto be nilpotent.Proposition 7.11 Let G be a nilpotent group. Then every proper subgroupof G is properly contained in its normaliser:H

Claim: P 1 P 2 ...P j∼ = P1 × P 2 ×···×P j for all j.Certainly this claim holds for j =1. Assumeitholdsforsomej, andconsider N = P 1 P 2 ...P j∼ = P1 ×···×P j G and P j+1 G. Then |N| iscoprime to |P j+1 |. Hence N ∩ P j+1 = 1 and therefore NP j+1 satisfies theconditions to be an (internal) direct product. ThusNP j+1∼ = N × Pj+1 ∼ = P1 × P 2 ×···×P j × P j+1 ,and by induction the claim holds.In particular, note|P 1 P 2 ...P k | = |P 1 × P 2 ×···×P k | = |P 1 |·|P 2 |·...·|P k | = |G|,soG = P 1 P 2 ...P k∼ = P1 × P 2 ×···×P k .(iii) ⇒ (i): Suppose G = P 1 ×P 2 ×···×P k ,adirectproductofnon-trivialp-groups. Then(by Corollary 2.41). ThenZ(G) =Z(P 1 ) × Z(P 2 ) ×···×Z(P k ) ≠ 1G/Z(G) =P 1 /Z(P 1 ) × P 2 /Z(P 2 ) ×···×P k /Z(P k )is a direct product of p-groups of smaller order. By induction, G/Z(G) isnilpotent, say γ c (G/Z(G)) = 1. Now apply Lemma 7.4(ii) to the naturalmap π : G → G/Z(G) toseethatγ c (G)π = γ c (G/Z(G)) = 1. Thusγ c (G) ker π =Z(G) andhenceγ c+1 (G) =[γ c (G),G] [Z(G),G]=1.Therefore G is nilpotent.□This tells us that the study of finite nilpotent groups reduces tounderstandingp-groups. We finish by introducing the Frattini subgroup, which isof significance in many parts of group theory.Definition 7.14 A maximal subgroup of a group G is a subgroup M

Lemma 7.15 Let G be a nilpotent group. Then every maximal subgroupof G is normal in G.□Definition 7.16 The Frattini subgroup Φ(G) ofagroupG is the intersectionof all its maximal subgroups:⋂Φ(G) = M.M maximalin G(If G is an (infinite) group with no maximal subgroups, then Φ(G) =G.)If we apply an automorphism to a maximal subgroup, we map it toanother maximal subgroup. Hence the automorphism group permutes themaximal subgroups of G.Lemma 7.17 If G is a group, then the Frattini subgroup Φ(G) is a characteristicsubgroup of G.□Our final theorem characterising nilpotent finite groups is:Theorem 7.18 Let G be a finite group. The following are equivalent:(i) G is nilpotent;(ii) H

(iv) ⇒ (v): Let P be a Sylow p-subgroup of G and let N = P Φ(G)(which is a subgroup of G, sinceΦ(G) G by Lemma 7.17). Let x ∈ Nand g ∈ G. Thenx −1 x g =[x, g] ∈ G ′ Φ(G) N.Hence x g ∈ N for all x ∈ N and g ∈ G, soN G. Now P is a Sylowp-subgroup of N (since it is the largest possible p-subgroup of G, so iscertainly largest amongst p-subgroups of N). Apply the Frattini Argument(Lemma 6.35):G =N G (P ) N=N G (P ) P Φ(G)=N G (P )Φ(G) (as P N G (P )).From this we deduce that G =N G (P ): for suppose N G (P ) ≠ G. ThenN G (P ) M < G for some maximal subgroup M of G. By definition,Φ(G) M, soN G (P )Φ(G) M

Thus S is a set of non-generators iffor all subsets X ⊆ G.G = 〈X, S〉 implies G = 〈X〉Lemma 7.21 The Frattini subgroup Φ(G) is a set of non-generators for afinite group G.Proof: Let G = 〈X, Φ(G)〉. If 〈X〉 ≠ G, then there exists a maximalsubgroup M of G such that 〈X〉 M