Maximal among abelian normal subgroups

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Definition

Symbol-free definition

A subgroup of a group is termed maximal among Abelian normal subgroups if it is an Abelian normal subgroup and there is no Abelian normal subgroup properly containing it.

Definition with symbols

A subgroup $H$ of a group $G$ is termed maximal among Abelian normal subgroups if $H$ is an Abelian normal subgroup of $G$, and for any $K$ containing $H$ that is an Abelian normal subgroup of $G$, $H = K$.

Formalisms

In terms of the maximal operator

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

Relation with other properties

Weaker properties

• Abelian normal subgroup

Related properties

• Maximal among Abelian characteristic subgroups
• Self-centralizing subgroup (if inside a supersolvable group): For full proof, refer: Maximal among Abelian normal implies self-centralizing in supersolvable