Maximal among abelian normal subgroups
From Groupprops
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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]
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 of a group
is termed maximal among Abelian normal subgroups if
is an Abelian normal subgroup of
, and for any
containing
that is an Abelian normal subgroup of
,
.
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
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