ON THE EXISTENCE OF G-PERMUTABLE SUBGROUPS IN SIMPLE SPORADIC GROUPS

A subgroup A of a group G is G-permutable in G if for every subgroup B <= G there is x is an element of G satisfying AB(x) = B-x A. A subgroup A is hereditarily G-permutable in G if A is E-permutable in every subgroup E of G which includes A. The Kourovka Notebook contains Problem 17.112: Which finite nonabelian simple groups G possess a proper (hereditarily) G-permutable subgroup? We answer the question for simple sporadic groups.