Finite groups with a certain number of elements pairwise generating a non-nilpotent subgroup

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Let n>0 be an integer and X be a class of groups. We say that a group G satisfies the condition (X,n) whenever in every subset with n+1 elements of G there exist distinct elements x,y such that <x,y> is in X. Let N and A be the classes of nilpotent groups and abelian groups, respectively. Here we prove that: (1) If G is a finite semi-simple group satisfying the condition (N,n), then |G|<c2[₂₁n]n² [₂₁n]!, for some constant c. (2) A finite insoluble group G satisfies the condition (N,21) if and only if G/Z*(G) A₅, the alternating group of degree 5, where Z*(G) is the hypercentre of G. (3) A finite non-nilpotent group G satisfies the condition (N, 4) if and only if G/Z*(G) S₃, the symmetric group of degree 3. (4) An insoluble group G satisfies the condition (A,21) if and only if G Z(G)× A₅, where Z(G) is the centre of G. (5) If d is the derived length of a soluble group satisfying the condition (A,n), then d=1 if n∈ {1,2} and d≤ 2n-3 if n≥ 2.