Central series

This article defines a property that can be evaluated for a subgroup series

View a complete list of properties of subgroup series

Definition

Central series of finite length

This is the default meaning of the term central series.

A subgroup series:

$G=K_{1}\geq K_{2}\geq \dots K_{c}\geq K_{c+1}=1$

is termed a central series if it satisfies the following conditions:

1. It is a normal series: every $K_{i}$ is normal in $G$
2. For every $i$, $K_{i}/K_{i+1}$ is contained in the center of $G/K_{i+1}$.

Equivalently, it should satisfy the condition that for every $i$:

$[G,K_{i}]\subset K_{i+1}$

Descending central series of possibly infinite or transfinite length

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Ascending central series of possibly infinite or transfinite length

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Equivalence of definitions

Further information: Equivalence of definitions of central series

Relation with other properties

Stronger properties

• Strongly central series

Weaker properties

• Normal series: For full proof, refer: Central series implies normal series
• Subnormal series