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Linear groups with many two‐generator soluble subgroups

Let G be a linear group of degree n over a field and let S be any generating set. It is proved that (a) if every pair of products of at most (n!)3n2 elements of S ∪ S− 1 generates a soluble group then G is soluble and (b) if G is soluble and every pair of products of at most n3+log2n elements of S ∪ S−1 generates a nilpotent group then G is locally nilpotent.