Requerimientos de un seguidor solar

The class of nilpotent groups is one of the important classes of finite non-abelian groups. In some sense, this class is the one closest to the class of abelian groups. Since any Sylow subgroups of a finite group are nilpotent, this class of groups appears frequently in group theory. From the view point of probability, almost all groups are nilpotent. For instance, among the groups of order at most 2000, more than 99% of them are nilpotent; see Besche, Eick and O’Brien (2002).

Definition 4.2.1 (1) A series of subgroups

G=K1 ≥K2 ≥ · · · ≥Ks+1 = 1

(starting from G and terminating in 1) is called a central series of G if [Ki, G]≤Ki+1, i= 1, . . . , s. The numbers is called the length of the series. By Proposition 4.1.2(5), we have KiEG for

alli and, by definition, we have Ki/Ki+1 ≤Z(G/Ki+1). (2) The group Gis called nilpotent if G has a central series.

If the lower central series of G

G=G1 ≥G2 ≥G3 ≥ · · · ≥Gn≥ · · ·, Gn= [G, G, . . . , Gn]

terminates in 1, then it is a central series ofGandGbecomes nilpotent. In particular, all abelian groups are nilpotent.

Definition 4.2.2 Let G be a group. The series

1 = Z0(G)≤Z1(G)≤ · · · ≤Zn(G)≤ · · ·

is called the upper central series of G if for each n, Zn(G)/Zn−1(G) is the center ofG/Zn−1(G).

Note that the groupZn(G) is the inverse image of the centerZ(G/Zn−1(G)) under the canonical projection G→ G/Zn−1(G), and hence all Zn(G)

Nilpotent groups 143

Lemma 4.2.1 Let G be a nilpotent group and let

G=K1 ≥K2 ≥ · · · ≥Ks+1 = 1 be a central series of G. Then

Gi ≤Ki ≤Z(s+1)−i(G) for all i= 1, . . . , s+ 1.

Proof: For the first inequality, use induction on i. If i = 1, K1 =

G=G1, the inequality holds. Now assume thati >1 andKi−1 ≥Gi−1. Then, Gi = [Gi−1, G]≤[Ki−1, G]≤Ki.

For the second inequality, use induction on j = (s+ 1)−i. Ifj = 0,

Ks+1 = 1 =Z0(G), the inequality holds. Now assume that j > 0 and

Ks+1−(j−1) ≤ Zj−1(G). We want to prove that Ks+1−j ≤ Zj(G) which

is equivalent to [Ks+1−j, G]≤Zj−1(G). Since [Ks+1−j, G]≤Ks+1−(j−1),

the latter holds. ¤

Theorem 4.2.2 LetG be a finite group. Then the following are equiv- alent:

(1) G is nilpotent;

(2) the lower central series of G terminates in 1; (3) the upper central series of G terminates in G.

If G is nilpotent, then both the lower and the upper central series of

G are central series in the sense of Definition 4.2.1 and they have the same length c = c(G), called the nilpotency class of G. Moreover, G

has no central series of length less than c.

Note that c(G) = 1 if and only if G is abelian. Also, by Theo- rem 4.2.2(3), the group having trivial center cannot be nilpotent.

Proof: (1)⇒(2) and (1)⇒(3): If G is nilpotent, there is a central series

G=K1 ≥K2 ≥ · · · ≥Ks+1 = 1.

By Lemma 4.2.1, Gi ≤ Ki ≤ Z(s+1)−i(G) for all i = 1, . . . , s+ 1.

respectively, and the lengths of both upper and lower central series are

≤s.

(2)⇒(1) and (3)⇒(1): Since [Gi, G] = Gi+1, if the lower central series terminates in 1, then it has finite length and hence is a central series in the sense of Definition 4.2.1. Also, by Definition 4.2.2, so is the upper central series.

Finally, since both the upper and the lower central series have the shortest length by Lemma 4.2.1, their lengths are equal. ¤

Since the center of any p-group is not trivial, one can show that all

p-groups are nilpotent. Moreover, one can easily verify the following theorem and hence the proof is omitted.

Theorem 4.2.3 (1) All subgroups and factor groups of a nilpotent group are nilpotent;

(2) The direct product of a finite number of nilpotent groups is nilpo- tent;

(3) All finite p-groups are nilpotent.

Theorem 4.2.4 LetG be a finite group. Then the following are equiv- alent:

(1) G is nilpotent;

(2) If H < G, then H < NG(H);

(3) Every maximal subgroupM of Gis normal, and hence |G:M|is a prime;

(4) Every Sylow subgroup of G is normal, and hence G is the direct product of its Sylow subgroups;

(5) Every Sylow subgroup of G is a characteristic subgroup.

Proof: (1)⇒(2): Let H < G. Then there exists a positive integer

i such

that Zi(G) ≤ H, but Zi+1(G) H. Choose z ∈ Zi+1(G)\H. By definition of Zi+1(G), we have z−1h−1zh = [z h] ∈ Zi(G) for

any h∈H which implies that z ∈ NG(H). Since z /∈H, we have

NG(H)> H.

(2) ⇒ (3): Let M be a maximal subgroup of G. By (2), M EG. Consider G = G/M. Since M is maximal, G has no nontrivial subgroup, hence |G|=|G:M| is a prime.

Nilpotent groups 145

(3) ⇒ (4): Let P be a Sylow p-subgroup of G and let H = NG(P).

If G 6=H, take a maximal subgroup M of G such that M ≥ H. By (3), we have MEG. But by Proposition 2.2.5, NG(M) =M,

a contradiction. Thus we have G = NG(P), that is P EG. It

follows that G is a direct product of its Sylow subgroups. (See Exercise 2.2.1.)

(4) ⇒(1): It is an immediate consequence of Theorem 4.2.3(2)-(3). (4) ⇒(5): See Exercise 2.2.7;

(5) ⇒(4) is clear. _¤

The Frobenius kernel of any Frobenius group is known to be nilpo- tent, but no Frobenius groups are nilpotent becauseNG(H) =Hfor the

Frobenius complementH.In particular, the groupFpq and the dihedral

group D2n with odd n are not nilpotent.

Corollary 4.2.5 Let G be nilpotent. Then for any divisor m of |G|, G has a subgroup of order m.

This corollary means that the converse of Lagrange Theorem holds for any nilpotent group.

Theorem 4.2.6 A nilpotent group G is solvable.

Proof: It is easy to prove by induction on i that Gi ≥G(i−1) for any

positive integer i. The conclusion follows. ¤

The converse of Theorem 4.2.6 does not hold in general. (The di- hedral groupD2n, n odd, is a counter example.)

Remark: A solvable group is a group having a normal series whose factor groups are all abelian, (see Example 3.1.1(1)). As a strengthen- ing of solvability, a group G is called supersolvable if it has a normal series whose factors are all cyclic. Hence, we can have the following arrangement of classes of groups:

cyclic ⇒ abelian ⇒nilpotent ⇒ supersolvable⇒ solvable⇒ finite group

Theorem 4.2.7 Let Gbe nilpotent and 16=NEG. Then [N, G]< N

andN∩Z(G)>1. In particular, every minimal normal subgroup of G

is contained in Z(G).

Proof: Set N1 =N. For i >1, we define Ni = [Ni−1, G]. Obviously,

Ni ≤ N and Ni ≤ Gi. Since G is nilpotent, there is an integer c such

that Gc+1 = 1. Hence Nc+1 = 1. It follows that N2 = [N, G] < N. (If not, for any i, we would have Ni = N, a contradiction.) Now assume

that t satisfies Nt = 1 and Nt−1 6= 1. Then by [Nt−1, G] = Nt = 1 we

have Nt−1 ≤ Z(G). On the other hand, Nt−1 ≤ N, and N ∩Z(G) ≥

Nt−1 6= 1. ¤

Theorem 4.2.8 (P. Hall) (1) Let

G=K1 ≥K2 ≥ · · · ≥Ks+1 = 1 (4.1) be any central series of a nilpotent group G. Then for anyi, j we have [Ki, Gj]≤Ki+j.

(2) For any i, j we have [Gi, Gj]≤ Gi+j and [Gi, Zj(G)]≤ Zj−i(G),

where we define Zj−i(G) = 1 whenj < i. In particular, for any i

we have [Gi, Zi(G)] = 1.

Proof: (1) Use induction on j. Assume j = 1. Since Eq. (4.1) is a central series of G, we have [Ki, G1] = [Ki, G] ≤ Ki+1 for all i, the conclusion holds. Now assumej >1. Since Gj = [Gj−1, G], we have

[Ki, Gj] = [Gj, Ki] = [Gj−1, G, Ki] for every i.

By the induction hypothesis,

[G, Ki, Gj−1] = [Ki, G, Gj−1]≤[Ki+1, Gj−1]

≤ Ki+1+j−1 =Ki+j,

[Ki, Gj−1, G] ≤ [Ki+j−1, G]≤Ki+j.

By Hall’s three subgroup Lemma, [Gj−1, G, Ki]≤Ki+j. It follows that

[Ki, Gj] ≤Ki+j.

(2) Replacing Eq. (4.1) by the lower and upper central series of G,

Frattini subgroups 147

Let G be a solvable group. The smallest number r = r(G) satis- fying G(r) _{= 1 is called the derived length of G. In Exercise 4.2.7, a} relation between the nilpotency class and the length of derived series of a nilpotent group is given.

Exercises

4.2.1. Prove Theorem 4.2.3.

4.2.2. Suppose that N and G/N are nilpotent. Is Galso nilpotent?

4.2.3. Let G=ha, bi be a metabelian group and let n≥2. ThenGn/Gn+1 can be generated byn−1 elements.

4.2.4. LetG/M and G/N be nilpotent. ThenG/M ∩N is nilpotent. 4.2.5. A finite groupGis nilpotent if and only if for anyx, y∈G, (o(x), o(y))

= 1 implies [x, y] = 1.

4.2.6. LetG be a group. SupposeG=G0_{. Then Z}

2(G) =Z1(G). 4.2.7. LetGbe a nilpotent group. ThenG(i) _≤G

2i,i= 0,1, . . . , r. It follows

that r(G) ≤ dlog₂(c+ 1)e, where dae is the smallest integer not less than the real number a.

4.2.8. Are the converses of Corollary 4.2.5 and Theorem 4.2.6 true? 4.2.9. Justify the following

(1) Nilpotent groups: abelian groups;p-groups; the qoaternion group Q8; every direct products of groups of prime power orders. (2) Non-nilpotent groups: any unsolvable groups in Exercise 2.3.5(2);

Frobenius groups; dihedral groups whose order is not a prime power;S3,A4,S4; and their direct products.