ON NONNILPOTENT SUBSETS IN GENERAL LINEAR GROUPS

Let G be a group. A subset X of G is said to be nonnilpotent if for any two distinct elements x and y in X, 〈x,y〉 is a nonnilpotent subgroup of G. If, for any other nonnilpotent subset X′ in G, ∣X∣≥∣X′ ∣, then X is said to be a maximal nonnilpotent subset and the cardinality of this subset is denoted by ω(𝒩G) . Using nilpotent nilpotentizers we find a lower bound for the cardinality of a maximal nonnilpotent subset of a finite group and apply this to the general linear group GL (n,q) . For all prime powers q we determine the cardinality of a maximal nonnilpotent subset of the projective special linear group PSL (2,q) , and we characterize all nonabelian finite simple groups G with ω(𝒩G)≤57 .