McLain groups over arbitrary rings and orderings

Abstract

In 1954, McLain [M] applied some well-known arguments of linear algebra on triangular matrices to establish the existence of characteristically simple, locally finitep-groups, now known as McLain groups [Rob, pp. 347–349]. His groups, having a trivial centre, illustrated sharply the difference between finite and locally finitep-groups. The construction of McLain groups depends on the dense linear ordering (ℚ,≤) and a fieldFpofpelements. It was immediately clear that the parametersFp, ℚ of the McLain groupG(Fp, ℚ) could be replaced by other linearly ordered, dense setsSand by other fieldsFwithout doing much harm to the construction. IfFhas characteristic 0, thenG(F, S) is still locally nilpotent but torsion-free. Wilson [W] investigatedG(Fp, S) for other orderings and Roseblade[R] deeply studied the automorphism group AutG(Fp, S) in his dissertation at Cambridge in 1963.