ON C-H-PERMUTABLE SUBGROUPS OF FINITE GROUPS

Let sigma = {sigma(i) vertical bar i is an element of I} be some partition of the set of all primes P, let G be a finite group, and sigma(G) = {sigma(i) vertical bar sigma(i) boolean AND pi(G) not equal empty set}. A set H of subgroups of G is a complete Hall sigma-set of G if every nonidentity member of H is a Hall sigma(i)-subgroup of G for some i is an element of I and H includes exactly one Hall sigma(i)-subgroup of G for every sigma(i) is an element of sigma(G). Let H be a complete Hall sigma-set of G and let C be a nonempty subset of G. We say that a subgroup H of G is C-H-permutable if for all A is an element of H there exists some x is an element of C such that H(x)A = AH(x). We investigate the structure of G by assuming that some subgroups of G are C-H-permutable. Some known results are generalized.