# 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**.

## References

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