Chief series
This article defines a property that can be evaluated for a subgroup seriesView a complete list of properties of subgroup series
Contents
Definition
Definition for finite length
Let be a group. A chief series for
is a subgroup series from
to the trivial subgroup where all members are normal subgroups of
, and where the series cannot be refined further. More explicitly:
- A series of subgroups:
is termed a chief series if are normal in
for all
,
is a proper subgroup of
, and there is no normal subgroup of
that properly contains
and is properly contained within
. In other words, the normal series cannot be refined further to another normal series.
- A series of subgroups:
is termed a chief series if each is normal in
and
is a minimal normal subgroup of
.
A group that possesses a chief series is termed a group of finite chief length. The factor groups for a chief series are termed the chief factors. It turns out that for a group of finite chief length, any two chief series have the same length and the lists of chief factors are the same.