Maximal among abelian subgroups

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This article describes a property that arises as the conjunction of a subgroup property: self-centralizing subgroup with a group property (itself viewed as a subgroup property): Abelian group
View a complete list of such conjunctions

Definition

Symbol-free definition

A subgroup of a group is termed maximal among abelian subgroups or a maximal abelian subgroup or a self-centralizing abelian subgroup if it satisfies the following equivalent conditions:

  1. It equals its centralizer in the whole group.
  2. It is abelian and self-centralizing.
  3. It is abelian and is not properly contained in a bigger abelian subgroup.

Definition with symbols

A subgroup H of a group G is termed maximal among Abelian subgroups or a maximal abelian subgroup or a self-centralizing abelian subgroup if it satisfies the following equivalent conditions:

  1. H = C_G(H), where C_G(H) denotes the centralizer of H in G.
  2. C(G)H \le H and H is abelian.
  3. H is abelian, and if H \le K \le G with K Abelian, then H = K.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of maximal among abelian subgroups

Formalisms

In terms of the maximal operator

This property is obtained by applying the maximal operator to the property: Abelian subgroup
View other properties obtained by applying the maximal operator

Relation with other properties

Stronger properties

Weaker properties

Facts