On nilpotent pi-subgroups in solvable groups

Let d(n, G) denote the maximum of the orders of all nilpotent subgroups wit h nilpotency class at most n of a finite group G. In particular d(infinity, G) means the maximum of the orders of all nilpotent subgroups of G. We prov e that if G is a finite solvable group whose Sylow 2-subgroups are of nilpo tency class c and H and K are pi-Hall and pi'-Hall subgroups of G respectiv ely then d(infinity, H) > d(2, K) implies O-pi(G) not equal 1 or O-sigma c( G) not equal 1, where sigma(c) is some finite set of primes depending only on c. This result is related to Burnside's other p(a)q(b) theorem.