Abelian normal subgroup of maximum order

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Definition

A subgroup $H$ of a group of prime power order $G$ is termed an abelian normal subgroup of maximum order if it is an abelian normal subgroup (specifically, an abelian normal subgroup of group of prime power order) and there is no abelian normal subgroup of $G$ of larger order than $H$.

Relation with other properties

Stronger properties

• Abelian subgroup of maximum order which is normal

Weaker properties

• Abelian normal subgroup of group of prime power order
• Abelian normal subgroup
• Maximal among abelian normal subgroups