A new notion of rank for finite supersolvable groups and free linear actions on products of spheres
Date
2003
Authors
Barker, L.
Yalçın, E.
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Source Title
Journal of Group Theory
Print ISSN
1433-5883
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Publisher
Walter de Gruyter GmbH
Volume
6
Issue
3
Pages
347 - 364
Language
English
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Abstract
For a finite supersolvable group G, we define the saw rank of G to be the minimum number of sections Gk/Gk-1 of a cyclic normal series G* such that Gk -Gk-1 contains an element of prime order. The axe rank of G, studied by Ray [10], is the minimum number of spheres in a product of spheres admitting a free linear action of G. Extending a question of Ray, we conjecture that the two ranks are equal. We prove the conjecture in some special cases, including that where the axe rank is 1 or 2. We also discuss some relations between our conjecture and some questions about Bieberbach groups and free actions on tori.