Solvable PST-groups, strong Sylow bases and mutually permutable products

AbstractA subgroup H of a group G is said to permute with the subgroup K of G if HK=KH. Subgroups H and K are mutually permutable (totally permutable) in G if every subgroup of H permutes with K and every subgroup of K permutes with H (if every subgroup of H permutes with every subgroup of K). If H and K are mutually permutable and H∩K=1, then H and K are totally permutable. A subgroup H of G is S-permutable in G if H permutes with every Sylow subgroup of G. A group G is called a PST-group if S-permutability is a transitive relation in G. Let {p1,…,pn,pn+1,…,pk} be the set of prime divisors of the order of a finite group G with {p1,…,pn} the set of prime divisors of the order of the normal subgroup N of G. A set of Sylow subgroups {P1,…,Pn,Pn+1,…,Pk}, Pi∈Sylpi(G), form a strong Sylow system with respect to N if PiPj is a mutually permutable product for all i∈{1,2,…,n} and j∈{1,2,…,k}. We show that a finite group G is a solvable PST-group if and only if it has a normal subgroup N such that G/N is nilpotent and G has a strong Sylow system with respect to N. It is also shown that G is a solvable PST-group if and only if G has a normal solvable PST-subgroup N and G/N″ is a solvable PST-group