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Locally finite minimal non-FC-groups

Published online by Cambridge University Press:  04 October 2011

Mahmut Kuzucuoglu  and
Richard E. Phillips
Mahmut Kuzucuoglu
Affiliation:
Department of Mathematics, Middle East Technical University, Ankara, Turkey
Richard E. Phillips
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A.
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Extract

We recall that a group G is an FC-group if for every x∈G the set of conjugates {xg|g∈G} is a finite set. Our interest here is with those groups G which are not FC groups while every proper subgroup of G is an FC-group: such groups are called minimal non-FC-groups. Locally finite minimal non-FC-groups with (G ≠ G′ are studied in [1] and the structure of these groups is reasonably well understood. In [2] Belyaev has shown that a perfect, locally finite, minimal non-FC-group is either a simple group or a p-group for some prime p. Here we make use of the results of [5] to refine the result of Belyaev and provide a positive answer to problem 5·1 of [11]; in particular, we prove the following

Theorem. There exists no simple, locally finite, minimal non-FC-group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 417 - 420
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

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