Abelian subgroup of maximum order

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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]
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

Related properties

Facts