On finite groups with σ-subnormal Schmidt subgroups

Let G be a finite group and sigma = {sigma(i)vertical bar i is an element of I} be some partition of the set of all primes. We say that G is: sigma-primary if G is a sigma(j)-group, forsome i; sigma-nilpotent if G = G(1) x ... x G(n) for some sigma-primary groups G(1),..., G(n). A sub group A of G is called sigma-subnormal in G if there is a subgroup chain A = A(0) <= A(1) <= ... <= A(m) = G such thateither A(i-1) (sic) A(i) or A(i)/(A(i-1))(Ai) is sigma-primary fo rall i = 1,..., m. In this paper we provethat the set of all sigma-subnormal (sigma-nilpotent) subgroups of G forms a sublattice of the lattice of all subgroups of G and, being based on this result, we prove also that if every Schmidt subgroup of G is sigma-subnormal in G, then the commutant subgroup G' is sigma-nilpotent.