Finite Groups with Given Systems of Generalized Subnormal and Generalized Sylow Permutable Subgroups

Throughout this paper, all groups are finite and G always denotes a finite group; L(G) is the lattice of all subgroups of G; sigma={sigma(i)|i is an element of I} is some partition of the set of all primes. A subgroup A of G is said to be: (i)sigma-subnormal in G if there is a subgroup chain A=A(0)<= A(1)<= <middle dot> <middle dot> <middle dot> <= A(n)=G such that either A(i-1)(sic)A(i) or A(i)/(A(i-1))A(i) is sigma-primary for all i= 1, . . . , n;(ii)sigma-permutableinGifGis sigma-full, that is, G has a Hall sigma i-subgroup for every sigma(i)is an element of sigma(G) and A permutes with all such Hall subgroups of G. In this review we discuss some new results on finite groups with given systems sigma-subnormal and sigma-permutable subgroups. In particular, we discuss groups inwhich sigma-permutability is a transitive relation in the group.