Abelian subgroup of maximum order

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This article is about a maximality notion among subgroups, related to abelianness or small class, in a group of prime power order.
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Definition

Let $P$ be a group of prime power order, i.e., a finite $p$-group for some prime $p$. A subgroup $A$ of $P$ is termed an abelian subgroup of maximum order (sometimes also a large abelian subgroup) if $A$ is an abelian subgroup, and the order of any abelian subgroup of $P$ divides the order of $A$.

The set of abelian subgroups of maximum order in a group $P$ of prime power order is sometimes denoted $\mathcal{A}(P)$. Their join, termed the join of abelian subgroups of maximum order, and is sometimes also termed the Thompson subgroup. This set of subgroups satisfies some powerful replacement theorems, most notably Thompson's replacement theorem.

Relation with other properties

Weaker properties

• Maximal among abelian subgroups

Related properties

• Abelian subgroup of maximum rank
• Elementary abelian subgroup of maximum order
• Subgroup with abelianization of maximum order
• Centrally large subgroup

Facts

• Two abelian subgroups of maximum order need not be isomorphic. For full proof, refer: Abelian subgroups of maximum order need not be isomorphic
• Two isomorphic abelian subgroups of maximum order need not be automorphic subgroups. For full proof, refer: Abelian of maximum order not implies isomorph-automorphic
• Two abelian subgroups of maximum order, that are automorphic, need not be conjugate. For full proof, refer: Abelian of maximum order not implies automorph-conjugate
• Not every abelian subgroup of a finite $p$-group is necessarily contained in an Abelian subgroup of maximum order. For full proof, refer: Abelian not implies contained in Abelian subgroup of maximum order