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