Ensuring a Group is Weakly Nilpotent

Abstract

Let m, n be positive integers and 𝔛 be a class of groups. We say that a group G satisfies the condition 𝔛(m, n), if for every two subsets M and N of cardinalities m and n, respectively, there exist x ∈ M and y ∈ N such that ⟨x, y⟩ ∈ 𝔛. In this article, we study groups G satisfies the condition 𝔑(m, n), where 𝔑 is the class of nilpotent groups. We conjecture that every infinite 𝔑(m, n)-group is weakly nilpotent (i.e., every two generated subgroup of G is nilpotent). We prove that if G is a finite non-soluble group satisfies the condition 𝔑(m, n), then , for some constant c (in fact c ≤ max{m, n}). We give a sufficient condition for solubility, by proving that a 𝔑(m, n)-group is a soluble group whenever m + n < 59. We also prove the bound 59 cannot be improved and indeed the equality for a non-soluble group G holds if and only if G ≅ A 5, the alternating group of degree 5.