Abelian subgroup of maximum rank

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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

Suppose $P$ is a group of prime power order. A subgroup $A$ of $P$ is termed an abelian subgroup of maximum rank if $A$ is an abelian subgroup of $P$ and the rank of $A$ (i.e., the minimum number of elements needed to generate $A$) is the maximum among the ranks of all abelian subgroups of $P$.

Note that abelian subgroups of maximum rank need not be maximal among abelian subgroups.

The join of all such subgroups is termed the join of abelian subgroups of maximum rank, and is one of the three Thompson subgroups often considered for groups of prime power order.

Relation with other properties

Stronger properties

• Maximal among abelian subgroups of maximum rank: These are the maximal elements with respect to inclusion among abelian subgroups of maximum rank.
• Elementary abelian subgroup of maximum order

Similar properties

• Abelian subgroup of maximum order