Nilpotent Groups

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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
Page 3: Note, however, thatN G, G/N and N
Page 8 and 9: Lemma 7.15 Let G be a nilpotent gro
Page 10: Thus S is a set of non-generators i

Nilpotent Groups

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