A Course on Group Theory by John S. Rose

Author:John S. Rose
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2013-06-26T16:00:00+00:00

is a proper normal series of G, which can therefore be refined to a chief series of G (by the version of 7.9 for groups with operators). Since the factors of the derived series are abelian, so are the factors of this chief series. Moreover, by the Jordan-Hölder theorem, every chief factor of G is isomorphic to one of the factors of this particular chief series. Thus all chief factors of G are abelian. By 7.38, they are also characteristically simple. Hence, by 7.41, they are elementary.

Suppose conversely that either every composition factor of G has prime order or every chief factor of G is elementary abelian. Then either a composition series or a chief series of G is an abelian series of G. Hence G is soluble.

Remarks. The theorem shows in particular that a finite soluble group has a series all of whose factors are cyclic. This is not true in general for infinite soluble groups: see 387.

A finite soluble group does not in general have a normal series all of whose factors are cyclic: 7.34 shows this. See also 389.

7.57. Suppose that G has a chief series. Any central chief factor of G is finite and has prime order.

Proof. In view of 7.36, it is enough to consider a minimal normal subgroup L of G such that L Z(G) and to show that | L | = p for some prime p. Since L Z(G) every subgroup of L is normal in G (118). Therefore, since L is minimal normal in G, the only subgroups of L are 1 and L. It follows (29) that | L | = p for some prime p.

7.58 Theorem. Let G be a finite group. Then the following two statements are equivalent:

(i) G is nilpotent.

(ii) Every chief factor of G is central

Proof. Suppose that G is nilpotent. Since quotient groups of nilpotent groups are nilpotent (7.46), it is enough, by 7.36, to prove that every minimal normal subgroup of G lies in the centre of G. Let L be a minimal normal subgroup of G. Then, by 3.51 and 3.53,

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