Abelian normal subgroup of maximum rank

Definition
A subgroup of a group of prime power order is termed an Abelian normal subgroup of maximum rank if it is an defining ingredient::Abelian normal subgroup (i.e., it is an defining ingredient::Abelian group and is a defining ingredient::normal subgroup) and its rank equals the maximum of the ranks of Abelian normal subgroups. Equivalently, it is an Abelian normal subgroup whose rank equals the normal rank of the whole group.

Stronger properties

 * Weaker than::Elementary Abelian normal subgroup of maximum rank

Facts

 * Abelian normal subgroup of maximum rank that is also maximal among Abelian normal subgroups exists
 * Omega-1 of maximal among Abelian normal subgroups with maximum rank in odd-order p-group equals omega-1 of centralizer