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In this paper, solubility of groups factorised as a product of two subgroups which are connected by certain permutability properties is studied. [ABSTRACT FROM AUTHOR]

A finite group G is said to be a -group if, for subgroups H and K of G with H Sylow-permutable in K and K Sylow-permutable in G, it is always the case that H is Sylowpermutable in G. A group G is a *-group if, for subgroups H and K of G with H normal in K and K normal in G, it is always the case that H is Sylow-permutable in G. In this paper, we show that the classes of finite -groups and finite *-groups coincide. A new characterization of soluble -groups is also presented. [ABSTRACT FROM AUTHOR]

In the paper, a Frattini-like subgroup associated with a class ?? of simple groups is introduced and analyzed. The corresponding ??-saturated formations are exactly the ??-local ones introduced by Förster. Our techniques are also useful to highlight the properties and behavior of ?-local formations. In fact, extensions and improvements of several results of Shemetkov are natural consequences of our study. [ABSTRACT FROM AUTHOR]

A characterisation of finite soluble groups in which Sylow permutability is a transitive relation by means of subgroup embedding properties enjoyed by all the subgroups is proved. The key point is an extension of a subnormality criterion due to Wielandt. [ABSTRACT FROM AUTHOR]

Two subgroups A and B of a group G are cosubnormal if A and B are subnormal in their join 〈A,B〉 and are strongly cosubnormal if every subgroup of A is cosubnormal with every subgroup of B. We find necessary and sufficient conditions for A and B to be strongly cosubnormal in 〈A,B〉 and, if Z is the hypercentre of G=〈A,B〉, we show that A and B are strongly cosubnormal if and only if G/Z is the direct product of AZ/Z and BZ/Z. We also show that projectors and residuals for certain formations can easily be constructed in such a group. Two subgroups A and B of a group G are N-connected if every cyclic subgroup of A is cosubnormal with every cyclic subgroup of B (N denotes the class of nilpotent groups). Though the concepts of strong cosubnormality and N-connectedness are clearly closely related, we give an example to show that they are not equivalent. We note, however, that if G is the product of the N-connected subgroups A and B, then A and B are strongly cosubnormal. [Copyright &y& Elsevier]