Locally (soluble-by-finite) groups with all proper insoluble subgroups of finite rank |
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Authors: | Martyn R Dixon Martin J Evans Howard Smith |
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Institution: | (1) Department of Mathematics, University of Alabama, 35487-0350 Tuscaloosa, AL., U.S.A.;(2) Howard Smith Department of Mathematics, Bucknell University, 17837 Lewisburg, PA., U.S.A. |
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Abstract: | A group G has finite rank r if every finitely generated subgroup of G is at most r-generator. If C is a class of groups then we let C* denote the class of groups G in which every proper subgroup of G is either of finite rank or in C. We let denote the class of soluble groups and the class of soluble groups of derived length at most d, where d is a positive integer. We let λ denote the set of closure operations
and let denote the λ-closure of the class of periodic locally graded groups. Amongst other results we prove that a soluble -group is either of finite rank or of derived length at most d and also that a group in the class is either locally soluble, or has finite rank, or is isomorphic to one of or for suitable locally finite fields .
The second author would like to thank the Department of Mathematics at Bucknell University for its hospitality while part
of this work was being done. |
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Keywords: | Mathematics Subject Classification (1991)" target="_blank">Mathematics Subject Classification (1991) 20F19 20E25 |
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