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Published online by Cambridge University Press: 20 April 2010
Abstract
A group G is called triply factorised if it can be written as G = A ⋉ M = B ⋉ M = AB for two subgroups A and B and a normal subgroup M of G. It is shown how such triply factorised groups can be constructed using nearrings. Moreover, if G = A ⋉ M = B ⋉ M = AB is any triply factorised group with A ∩ B = 1, there exists a nearring by which the group G can be constructed. Finally, some structural properties of nearrings are described.
Introduction
A group G is called factorised if G = AB is the product of two subgroups A and B of G. If, in addition, there exists a normal subgroup M of G such that G = A ⋉ M = B ⋉ M = AB is a semidirect product of A resp. B and M, then G is called triply factorised by A, B, and M. In this case, A and B are complements of M and hence isomorphic.
Examples 1.1
Let G be an arbitrary group, and let A = B = G and M = 1. Then G is triply factorised by A, B, and M. In the following these trivial triply factorisations will not be considered.
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