Central series
From Groupprops
This article defines a property that can be evaluated for a subgroup seriesView a complete list of properties of subgroup series
Definition
Central series of finite length
This is the default meaning of the term central series.
is termed a central series if it satisfies the following conditions:
- It is a normal series: every
is normal in
- For every
,
is contained in the center of
.
Equivalently, it should satisfy the condition that for every :
Descending central series of possibly infinite or transfinite length
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Ascending central series of possibly infinite or transfinite length
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Equivalence of definitions
Further information: Equivalence of definitions of central series
Relation with other properties
Stronger properties
Weaker properties
- Normal series: For full proof, refer: Central series implies normal series
- Subnormal series