Central series

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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 \ge K_2 \ge \dots K_c \ge 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

Weaker properties