Maximal among abelian subgroups

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