The theory of formations of soluble groups, developed by Gaschütz [4], Carter and Hawkes[1], provides fairly general methods for investigating canonical full conjugate sets of subgroups in finite, soluble groups. Those methods, however, cannot be applied to the class of all finite groups, since strong use was made of the Theorem of Galois on primitive soluble groups. Nevertheless, there is a possiblity to extend the results of the above mentioned papers to the case of Π-soluble groups as defined by Čunihin [2]. A finite group G is called Π-soluble, if, for a given set it of primes, the indices of a composition series of G are either primes belonging to It or they are not divisible by any prime of Π In this paper, we shall frequently use the following result of Čunihin [2]: Ift is a non-empty set of primes, Π′ its complement in the set of all primes, and G is a Π-soluble group, then there always exist Hall Π-subgroups and Hall ′-subgroups, constituting single conjugate sets of subgroups of G respectively, each It-subgroup of G contained in a Hall Π-subgroup of G where each ′-subgroup of G is contained in a Hall Π′-subgroup of G. All groups considered in this paper are assumed to be finite and Π-soluble. A Hall Π-subgroup of a group G will be denoted by G.
Finite groups with permutable Hall subgroups
