Ensuring a finite group is supersoluble

Abstract

A special case of the main result is the following. Let G be a finite, non-supersoluble group in which from arbitrary subsets X, Y of cardinality n we can always find x € X and y 6 7 generating a supersoluble subgroup. Then the order of G is bounded by a function of n. This result is a finite version of one line of development of B.H. Neumann's well-known and much generalised result of 1976 on infinite groups.

Ensuring a Group is Weakly Nilpotent

“…At the end of this section, we note that Bryce in [6], defines a class we might write as m n -groups, whenever m = n. He studied questions similar to Neumann's in which, instead of the property of being abelian, he considers other properties of groups, such as being supersoluble and nilpotency. He answers the questions by finding some functions with properties similar to those of the function f m n in the abelian case.…”

“…Suppose that G is a finite non-soluble m n -group. Then a tidier bound for G can be obtained by Remark 2.3, Proposition 4 of [6], and Theorem 3.4, as follows:…”