A criterion for a finite group to be σ-soluble

Let sigma ={sigma(i)vertical bar i is an element of l} 1/4 friji 2 Ig be a partition of the set of all primes P and G a finite group. G is said to be sigma-soluble if every chief factor H/K of G is sigma-primary (that is, H/K is a sigma(i)-group for some i = i(H/K)). A subgroup A of G is called 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))(Ai) is sigma-primary for all i = 1, ...,n. Denote by i(sigma)(G) the number of classes of iso-ordic non-sigma-subnormal sub-groups of G. In this note, we study the structure of G depending on the invariant i(sigma)(G). In particular, the following criterion is proved. Theorem 1.2. If i(sigma)(G) <= 2 vertical bar sigma(G)vertical bar, then G is sigma-soluble.