ON THE EXISTENCE OF HEREDITARILY G-PERMUTABLE SUBGROUPS IN EXCEPTIONAL GROUPS G OF LIE TYPE

A subgroup A of a group G is G-permutable in G if for every subgroup B <= G there exists x is an element of G such that 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 has Problem 17.112(b): Which finite nonabelian simple groups G possess a proper hereditarily G-permutable subgroup? We answer this question for the exceptional groups of Lie type. Moreover, for the Suzuki groups G congruent to B-2(2)(q) we prove that a proper subgroup of G is G-permutable if and only if the order of the subgroup is 2. In particular, we obtain an infinite series of groups with G-permutable subgroups.