Hall Subgroups and π-Separable Groups Original Research Article

LetGbe a finite group and π a set of primes. In this paper, some new criteria for π-separable groups and π-solvable groups in terms of Hall subgroups are proved. The main results are the following: Theorem. G is aπ-separable group if and only if (1) G satisfies Eπand Eπ′; (2) G satisfies Eπ union or logical sum {q}and Eπ′ union or logical sum {p}for all p set membership, variant π,q set membership, variant π′. Theorem. G is aπ-separable group if and only if (1) G satisfies Eπand Eπ′; (2) G satisfies Ep, qfor all p set membership, variant πand q set membership, variant π′.