On the intersection of a class of maximal subgroups of a finite group II

AbstractGiven a finite group G and any set of primes π, we define here two subgroups Sπ(G) and Φπ(G) which are generalisations of the Frattini subgroup of G. (If π′ denotes the complimentary set of primes, then we also have the corresponding subgroups where π is replaced by π′ in the definition.) When G is π-solvable, the results proved here include: (i) G is solvable if and only if both GSπ(G) and GSπ′(G) are solvable, (ii) Sπ(G) is supersolvable if and only if Sπ(G)Φ(G) is supersolvable, assuming Sπ(G) is solvable, (iii) if π = 2, then Φπ(G) is solvable