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Locally Finite Groups with All Subgroups Normal-by-(Finite Rank)

✍ Scribed by E.I Khukhro; H Smith

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 194 KB
Volume: 200
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1997.7264

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✦ Synopsis

A group is said to have finite special rank F s if all of its finitely generated subgroups can be generated by s elements. Let G be a locally finite group and suppose that HrH has finite rank for all subgroups H of G, where H denotes

the normal core of H in G. We prove that then G has an abelian normal subgroup Ž . whose quotient is of finite rank Theorem 5 . If, in addition, there is a finite number r bounding all of the ranks of HrH , then G has an abelian subgroup G Ž . whose quotient is of finite rank bounded in terms of r only Theorem 4 . These results are based on analogous theorems on locally finite p-groups, in which case Ž . the group G is also abelian-by-finite Theorems 2 and 3 .

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