Supersolubility and Finitary Groups

Supersolubility and Finitary Groups

In 2000, I completed a PhD in Finitary Group Theory under the supervision of Prof B.A.F. Wehrfritz. My examiners were Dr Peter M Neumann, O.B.E. and Prof Paul Flavell.

Abstract

We generalize Platonov’s theorems on linear groups of finite Prüfer rank and linear groups satisfying non-trivial laws to certain finitary skew linear groups. We show that there are no transitive finitary permutation groups of infinite degree with finite Prüfer rank. We also show that an irreducible finitary linear group of infinite dimension has infinite Prüfer rank and generates the variety of all groups.

We show that a locally supersoluble group that is either a transitive finitary permutation group on an infinite set or an irreducible finitary skew linear group of infinite dimension, is a p-group for some suitably chosen prime p.

We show that a locally-nilpotent by Abelian group that is either a transitive finitary permutation group on an infinite set or an irreducible finitary skew linear group of infinite dimension is a p-group for some suitably chosen prime p.

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