Chief series

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This article defines a property that can be evaluated for a subgroup series


View a complete list of properties of subgroup series

Definition

Definition for finite length

Let G be a group. A chief series for G is a subgroup series from G to the trivial subgroup where all members are normal subgroups of G, and where the series cannot be refined further. More explicitly:

  • A series of subgroups:

G = N_0 \ge N_1 \ge \dots \ge N_r = \{ e \}

is termed a chief series if N_i are normal in G for all i, N_{i+1} is a proper subgroup of N_i, and there is no normal subgroup of G that properly contains N_{i+1} and is properly contained within N_i. In other words, the normal series cannot be refined further to another normal series.

  • A series of subgroups:

G = N_0 \ge N_1 \ge \dots \ge N_r = \{ e \}

is termed a chief series if each N_i is normal in G and N_{i-1}/N_i is a minimal normal subgroup of G/N_i.

A group that possesses a chief series is termed a group of finite chief length. The factor groups for a chief series are termed the chief factors. It turns out that for a group of finite chief length, any two chief series have the same length and the lists of chief factors are the same.

Relation with other properties

Weaker properties

Related properties