Chief series

Last updated June 23, 2023

In abstract algebra, a chief series is a maximal normal series for a group.

Definition

A chief series is a maximal normal series for a group. Equivalently, a chief series is a composition series of the group G under the action of inner automorphisms.

In detail, if G is a group, then a chief series of G is a finite collection of normal subgroups N_i ⊆ G,

1=N_{0}\subseteq N_{1}\subseteq N_{2}\subseteq \cdots \subseteq N_{n}=G,

such that each quotient group N_i+1/N_i, for i = 1, 2,..., n − 1, is a minimal normal subgroup of G/N_i. Equivalently, there does not exist any subgroup A normal in G such that N_i<A<N_i+1 for any i. In other words, a chief series may be thought of as "full" in the sense that no normal subgroup of G may be added to it.

The factor groups N_i+1/N_i in a chief series are called the chief factors of the series. Unlike composition factors, chief factors are not necessarily simple. That is, there may exist a subgroup A normal in N_i+1 with N_i<A<N_i+1, but A is not normal in G. However, the chief factors are always characteristically simple, that is, they have no proper nontrivial characteristic subgroups. In particular, a finite chief factor is a direct product of isomorphic simple groups.

Properties

Existence

Finite groups always have a chief series, though infinite groups need not have a chief series. For example, the group of integers Z with addition as the operation does not have a chief series. To see this, note Z is cyclic and abelian, and so all of its subgroups are normal and cyclic as well. Supposing there exists a chief series N_i leads to an immediate contradiction: N₁ is cyclic and thus is generated by some integer a, however the subgroup generated by 2a is a nontrivial normal subgroup properly contained in N₁, contradicting the definition of a chief series.

Uniqueness

When a chief series for a group exists, it is generally not unique. However, a form of the Jordan–Hölder theorem states that the chief factors of a group are unique up to isomorphism, independent of the particular chief series they are constructed from^[1] In particular, the number of chief factors is an invariant of the group G, as well as the isomorphism classes of the chief factors and their multiplicities.

Other properties

In abelian groups, chief series and composition series are identical, as all subgroups are normal.

Given any normal subgroup N ⊆ G, one can always find a chief series in which N is one of the elements (assuming a chief series for G exists in the first place.) Also, if G has a chief series and N is normal in G, then both N and G/N have chief series. The converse also holds: if N is normal in G and both N and G/N have chief series, G has a chief series as well.

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References

↑ Lafuente, J. (November 1978). "Homomorphs and formations of given derived class". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press. 84 (3): 437–442. doi:10.1017/S0305004100055262.

Isaacs, I. Martin (1994). Algebra: A Graduate Course. Brooks/Cole. ISBN 0-534-19002-2.

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[Lemma-1] Lafuente, J. (November 1978). "Homomorphs and formations of given derived class". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press. 84 (3): 437–442. doi:10.1017/S0305004100055262.

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