On weakly sigma-permutable subgroups of finite groups

Let G be a finite group and {\sigma} = {{\sigma}_i, i \in I} be a partition of the set of all primes \mathbb{P}. A set \mathcal{H} of subgroups of G with 1 \in \mathcal{H} is said to be a complete Hall {\sigma}-set of G if every non-identity member of \mathcal{H} is a Hall {\sigma}_i-subgroup of G. A subgroup H of G is said to be {\sigma}-permutable if G possesses a complete Hall {\sigma}-set \mathcal{H} such that HA^x = A^xH for all A \in \mathcal{H} and all x \in G. We say that a subgroup H of G is weakly {\sigma}-permutable in G if there exists a {\sigma}-subnormal subgroup T of G such that G = HT and H \cap T \leq H_{\sigma}G. where H_{\sigma}G is the subgroup of H generated by all those subgroups of H which are {\sigma}-permutable in G. By using this new notion, we establish some new criterias for a group G to be a {\sigma}-soluble and supersoluble, and also we give the conditions under which a normal subgroup of G is hypercyclically embedded.