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On supplements of subgroups of finite groups

Xianhua LiA. Ballester-Bolinches — 2006

Bollettino dell'Unione Matematica Italiana

In this paper the concept of s-pair for a subgroup of a finite group is introduced and studied. It provides a uniform way to study the influence of some families of subgroups on the structure of a finite group. A criterion for a finite group to belong to a saturated formation and necessary and sufficient conditions for solubility, supersolvability and nilpotence of a finite group are given.

On some permutable products of supersoluble groups.

Manuel J. AlejandreA. Ballester-BolinchesJohn CosseyM. C. Pedraza-Aguilera — 2004

Revista Matemática Iberoamericana

It is well known that a group G = AB which is the product of two supersoluble subgroups A and B is not supersoluble in general. Under suitable permutability conditions on A and B, we show that for any minimal normal subgroup N both AN and BN are supersoluble. We then exploit this to establish some sufficient conditions for G to be supersoluble.

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