Series (group theory)

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In group theory, a series is a chain (mathematics) of subgroups of a group ordered by subset inclusion. The structure of the group is closely related to the existence of series with particular properties.

A series is a linearly ordered chain of subgroups of a given group G beginning with the group G itself:

${\displaystyle G=A_{0}\supseteq A_{1}\supseteq \cdots \supseteq A_{n}.\,}$

If the final group in the series is H we speak of a series from G to H.

The series is subinvariant or subnormal if each subgroup is a normal subgroup of its predecessor, ${\displaystyle A_{i}\triangleleft A_{i-1}}$. A subinvariant series in which each subgroup is a maximal normal subgroup of its predecessor is a composition series.

The series is invariant or normal if each subgroup is a normal subgroup of the whole group. A subinvariant series in which each subgroup is a normal subgroup of G maximal subject to being a proper subgroup of its predecessor is a principal series or chief series.

References[edit]

• Marshall Hall jr (1959). The theory of groups. New York: Macmillan, 123-124.