Chief series

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

It is similar to a composition series, though the two concepts are distinct in general: a chief series is a maximal normal series, while a composition series is a maximal subnormal series.

Chief series can be thought of as breaking the group down into less complicated pieces, which may be used to characterize various qualities of the group.

Definition[edit]

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,

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

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[edit]

Existence[edit]

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[edit]

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[edit]

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.

References[edit]

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