C-closed self-centralizing subgroup

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: c-closed subgroup and self-centralizing subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

Symbol-free definition

A subgroup of a group is termed a c-closed self-centralizing subgroup or a centralizer of Abelian subgroup if it satisfies the following equivalent conditions:

  1. Its centralizer equals its center and it equals the centralizer of its center (all relative to the whole group).
  2. It is self-centralizing (i.e., it contains its own centralizer in the whole group) and also occurs as the centralizer of some subgroup of the whole group.
  3. It occurs as the centralizer of some Abelian subgroup of the whole group.

Definition with symbols

A subgroup of a group is termed a c-closed self-centralizing subgroup or a centralizer of Abelian subgroup if it satisfies the following equivalent conditions:

  1. and .
  2. and there exists a subgroup such that .
  3. There exists an Abelian subgroup of such that .

Relation with other properties

Stronger properties

Weaker properties