On σ-Subnormal Subgroups of Finite Groups

Throughout this paper, all groups are finite and sigma is some partition of the set of all primes P (that is, sigma = {sigma(i) vertical bar i is an element of I}, where P = boolean OR(i is an element of I)sigma(i) and sigma(i) boolean AND sigma(j) = empty set for all i not equal j). A subgroup A of a group G is said to be sigma-subnormal in G if there is a subgroup chain A = A(0) <= A(1) <= . . . <= A(n) = G such that either A(i-1) >= A(i) or A(i)/(A(i-1))A(i) is a sigma(j)-group for some j = j(i) for all i = 1, . . ., n. In this review, we discuss some known results of the theory of sigma-subnormal subgroups and also some open questions in this line research.