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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.

A new notion of rank for finite supersolvable groups and free linear actions on products of spheres