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:

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

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, $A_{i}\triangleleft A_{i-1}$. 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

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