chief series
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In
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, a chief series is a maximal
normal series In mathematics, specifically group theory, a subgroup series of a group (mathematics), group G is a Chain (order theory), chain of subgroups: :1 = A_0 \leq A_1 \leq \cdots \leq A_n = G where 1 is the trivial group, trivial subgroup. Subgroup series ...
for a
group A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ident ...
. It is similar to a
composition series In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
, 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

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 automorphism In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s. In detail, if ''G'' is a
group A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ident ...
, then a chief series of ''G'' is a finite collection of
normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s ''N''''i'' ⊆ ''G'', :1=N_0\subseteq N_1\subseteq N_2\subseteq\cdots\subseteq N_n=G, such that each
quotient group A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...
''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 group, 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 group, characteristically simple, that is, they have no proper nontrivial characteristic subgroups. In particular, a finite chief factor is a direct product of groups, 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 group, cyclic and abelian group, 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: ''N''1 is cyclic and thus is generated by some integer ''a'', however the subgroup generated by 2''a'' is a nontrivial normal subgroup properly contained in ''N''1, 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. In particular, the number of chief factors is an invariant (mathematics), invariant of the group ''G'', as well as the isomorphism class, 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.


References

* {{cite book , last=Isaacs , first=I. Martin , authorlink=Martin Isaacs, title=Algebra: A Graduate Course , publisher=Brooks/Cole , year=1994 , isbn=0-534-19002-2 Subgroup series