Abelian and nilpotent subgroups of maximal order of groups of odd order

Denote the maximum of the orders of all nilpotent subgroups A of class at most c, of a finite group G, by d_c(G). LetA_c (G) be the set of all nilpotent subgroups of class at most c andhaving order d_c(G) in G. Let A_∞(G) denote the set of allnilpotent subgroups of maximal order of a group G.The aim of this paper is to investigate the set A_∞(G) ofgroups G of odd order and the structure of the groups G withthe property A₂(G) ⊆ A_∞(G). Theorem 1 gives an expressionfor the number of elements in A_∞(G). Theorem 2 gives criteriafor the nilpotency of groups of odd order.