Maximal among abelian subgroups

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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

• In a nilpotent group or supersolvable group, maximal among abelian normal subgroups: Further information: Maximal among abelian normal implies self-centralizing in nilpotent, maximal among abelian normal implies self-centralizing in supersolvable

Weaker properties

• c-closed self-centralizing subgroup
• c-closed subgroup
• Self-centralizing subgroup
• Subgroup containing the center

Facts

• In any group, there always exist maximal among abelian subgroups. In fact, every abelian subgroup of a group is contained in a maximal among abelian subgroup. Further information: Every abelian subgroup is contained in a maximal among abelian subgroups
• Two subgroups that are maximal among abelian subgroups need not be isomorphic. In fact, they may not even have the same size. For instance, in the symmetric group on three letters, there is a subgroup of order two and a subgroup of order three, both of them maximal among abelian subgroups.
• In fact, any group can be expressed as a union of subgroups that are maximal among abelian subgroups. In particular, any non-abelian group has at least three distinct maximal among abelian subgroups. Further information: Every group is a union of maximal among abelian subgroups
• In certain cases, any abelian subgroup can be replaced by a normal subgroup or 2-subnormal subgroup of the same size. Further information: Category:Replacement theorems