Trifactorisable groups

E-ISSN： 1755-1633|8|3|461-469

ISSN： 0004-9727

Source： Bulletin of the Australian Mathematical Society, Vol.8, Iss.3, 1973-06, pp. : 461-469

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

The group G is called trifactorisable if G has three subgroups, A, B, and C such that G = AB = BC = CA. Obviously the structure of the group G will be restricted by the structure of these subgroups. In this paper it will be shown that a finite group G is π-separable if and only if it satisfies Dπ and has a trifactorisation with two factors π closed and the third, C say, π-separable. In this case we show that the π- and π^′-lengths of G can be at most one more than those of C, and so it is this factor which “controls” the structure of G. Similar results are proved for π-solubility and solubility.