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 Ni⊆G,
such that each quotient group Ni+1/Ni, for i = 1, 2,..., n − 1, is a minimal normal subgroup of G/Ni. Equivalently, there does not exist any subgroup A normal in G such that Ni < A < Ni+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 Ni+1/Ni 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 Ni+1 with Ni < A < Ni+1 but A is not normal in G. However, the chief factors are always characteristically simple, that is, they have no non-identity proper characteristic subgroups. In particular, a finite chief factor is a direct product of isomorphic simple groups.
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