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McLain groups over arbitrary rings and orderings

Published online by Cambridge University Press:  24 October 2008

Manfred Droste  and
Rüdiger Göbel
Manfred Droste
Affiliation:
Institut für Algebra, Technische Universität Dresden, 01062 Dresden, Germany
Rüdiger Göbel
Affiliation:
FB 6, Mathematik und Informatik, Universität GHS Essen, 45117 Essen, Germany
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Extract

In 1954, McLain [M] applied some well-known arguments of linear algebra on triangular matrices to establish the existence of characteristically simple, locally finite p-groups, now known as McLain groups [Rob, pp. 347–349]. His groups, having a trivial centre, illustrated sharply the difference between finite and locally finite p-groups. The construction of McLain groups depends on the dense linear ordering (ℚ,≤) and a field Fp of p elements. It was immediately clear that the parameters Fp, ℚ of the McLain group G(Fp, ℚ) could be replaced by other linearly ordered, dense sets S and by other fields F without doing much harm to the construction. If F has characteristic 0, then G(F, S) is still locally nilpotent but torsion-free. Wilson [W] investigated G(Fp, S) for other orderings and Roseblade[R] deeply studied the automorphism group Aut G(Fp, S) in his dissertation at Cambridge in 1963.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 3 , May 1995 , pp. 439 - 467
Copyright
Copyright © Cambridge Philosophical Society 1995

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