A new notion of rank for finite supersolvable groups and free linear actions on products of spheres
نویسنده
چکیده
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 owns 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. 2000 Mathematics Subject Classification. Primary: 20D15; Secondary: 20J05, 57S25.
منابع مشابه
Free Actions of Finite Groups on S × S
Let p be an odd prime. We construct a non-abelian extension Γ of S1 by Z/p × Z/p, and prove that any finite subgroup of Γ acts freely and smoothly on S2p−1 × S2p−1. In particular, for each odd prime p we obtain free smooth actions of infinitely many non-metacyclic rank two p-groups on S2p−1 × S2p−1. These results arise from a general approach to the existence problem for finite group actions on...
متن کاملA New Finite Element Formulation for Buckling and Free Vibration Analysis of Timoshenko Beams on Variable Elastic Foundation
In this study, the buckling and free vibration of Timoshenko beams resting on variable elastic foundation analyzed by means of a new finite element formulation. The Winkler model has been applied for elastic foundation. A two-node element with four degrees of freedom is suggested for finite element formulation. Displacement and rotational fields are approximated by cubic and quadratic polynomia...
متن کاملExamples of Free Actions on Products of Spheres
We construct a non-abelian extension Γ of S by Z/3 × Z/3, and prove that Γ acts freely and smoothly on S × S. This gives new actions on S × S for an infinite family P of finite 3-groups. We also show that any finite odd order subgroup of the exceptional Lie group G2 admits a free smooth action on S × S. This gives new actions on S×S for an infinite family E of finite groups. We explain the sign...
متن کاملTriple factorization of non-abelian groups by two maximal subgroups
The triple factorization of a group $G$ has been studied recently showing that $G=ABA$ for some proper subgroups $A$ and $B$ of $G$, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. In this paper we study two infinite classes of non-abelian finite groups $D_{2n}$ and $PSL(2,2^{n})$...
متن کاملFusion Systems and Constructing Free Actions on Products of Spheres
We show that every rank two p-group acts freely and smoothly on a product of two spheres. This follows from a more general construction: given a smooth action of a finite group G on a manifold M , we construct a smooth free action on M×S1×· · ·×Sk when the set of isotropy subgroups of the G-action on M can be associated to a fusion system satisfying certain properties. Another consequence of th...
متن کامل