A Generalization of σ-Permutability

Throughout this paper,all groups are finite and G always denotes a finite group;σis some partition of the set of all primes P.A group G is said to beσ-primary if G is aπ-group for someπ∈σ.Aπ-semiprojector of G[29]is a subgroup H of G such that HN/N is a maximalπ-subgroup of G/N for all normal subgroups N of G.LetП⊆σ.Then we say thatχ={X_(1),...,X_(t)}is aП-covering subgroup system for a subgroup H in G if all members of the setχareσ-primary subgroups of G and for eachπ∈Пwithπ∩π(H)≠φthere are an index i and aπ-semiprojector U of H such that U≤X_(i).We study the embedding properties of subgroups H of G under the hypothesis that G has aП-covering subgroup systemχsuch that H permutes with X^(x)for all X∈χand x∈G.Some well-known results are generalized.