Group whose chief series are composition series
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A group whose chief series are composition series is a group that satisfies the following equivalent conditions:
- The group has finite composition length, and finite chief length, and the two lengths are equal.
- The group has a chief series of finite length that is also a composition series.
- The group has a chief series of finite length, and every chief series for the group is a composition series.
Note that it is not necessary for such a group that every composition series is a chief series. In fact, every composition series is a chief series if and only if the group is a T-group having finite composition length, i.e., it has finite composition length, and every subnormal subgroup is normal. This is because any subnormal series can be refined to a composition series for a group of finite composition length.