On the subgroup permutability degree of some finite simple groups.

Abstract

Consider a finite group G and subgroups H;K of G. We say that H and K permute if HK = KH and call H a permutable subgroup if H permutes with every subgroup of G. A group G is called quasi-Dedekind if all subgroups of G are permutable. We can define, for every finite group G, an arithmetic quantity that measures the probability that two subgroups (chosen uniformly at random with replacement) permute and we call this measure the subgroup permutability degree of G. This measure quantifies, among others, how close a finite group is to being quasi-Dedekind, or, equivalently, nilpotent with modular subgroup lattice. The main body of this thesis is concerned with the behaviour of the subgroup permutability degree of the two families of finite simple groups PSL2(2n), and Sz(q). In both cases the subgroups of the two families of simple groups are completely known and we shall use this fact to establish that the subgroup permutability degree in each case vanishes asymptotically as n or q respectively tends to infinity. The final chapter of the thesis deviates from the main line to examine groups, called F-groups, which behave like nilpotent groups with respect to the Frattini subgroup of quotients. Finally, we present in the Appendix joint research on the distribution of the density of maximal order elements in general linear groups and offer code for computations in GAP related to permutability