Series (group theory)

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:


 * $$G = A_0 \supseteq A_1 \supseteq \cdots \supseteq A_n . \, $$

The series is subinvariant or subnormal if each subgroup is a normal subgroup of its predecessor, $$A_i \triangleleft A_{i-1}$$. 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.