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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 simple pieces which may be used to characterize various qualities of the 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'', : 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 non-identity proper characteristic subgroups. In particular, a finite chief factor is a direct product of isomorphic simple groups. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Chief series」の詳細全文を読む スポンサード リンク
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