On solvable factors of almost simple groups

Let G be a finite almost simple group with socle G0. A (nontrivial) factorization of is an expression the form G=HK, where factors H and K are core-free subgroups. There extensive literature on factorizations groups, important applications in permutation theory algebraic graph theory. In recent paper, Xia second-named author describe groups solvable factor H. Several infinite families arise context classical each case subgroup G0 containing H?G0 identified. Building this earlier work, paper we compute sharp lower bound order every determine exact factor. As application, primitive nilpotent regular subgroup, extending results Burnside Schur cyclic subgroups, more work Li abelian case.