Some restrictions on finite groups acting freely on

Author:
Gunnar Carlsson

Journal:
Trans. Amer. Math. Soc. **264** (1981), 449-457

MSC:
Primary 55M35; Secondary 55S10

MathSciNet review:
603774

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Abstract: Restrictions other than rank conditions on elementary abelian subgroups are found for finite groups acting freely on , with trivial action on homology. It is shown that elements of order , an odd prime, with in the normalizer of an elementary abelian -subgroup of , must act trivially on unless . It is also shown that if or , must act trivially, independent of . This produces a large family of groups which do not act freely on for any values of and . For certain primes , using the mod two Steenrod algebra, one may show that acts trivially unless , where is an integer depending on .

**[1]**J. F. Adams and C. W. Wilkerson,*Finite 𝐻-spaces and algebras over the Steenrod algebra*, Ann. of Math. (2)**111**(1980), no. 1, 95–143. MR**558398**, 10.2307/1971218**[2]**Gunnar Carlsson,*On the nonexistence of free actions of elementary abelian groups on products of spheres*, Amer. J. Math.**102**(1980), no. 6, 1147–1157. MR**595008**, 10.2307/2374182**[3]**P. E. Conner,*On the action of a finite group on 𝑆ⁿ×𝑆ⁿ*, Ann. of Math. (2)**66**(1957), 586–588. MR**0096235****[4]**John Milnor,*Groups which act on 𝑆ⁿ without fixed points*, Amer. J. Math.**79**(1957), 623–630. MR**0090056****[5]**R. Oliver, Proc. Aarhus Sympos. on Algebraic Topology, 1978, Lecture Notes in Math., vol. 763, Springer-Verlag, Berlin and New York.**[6]**Richard P. Stanley,*Invariants of finite groups and their applications to combinatorics*, Bull. Amer. Math. Soc. (N.S.)**1**(1979), no. 3, 475–511. MR**526968**, 10.1090/S0273-0979-1979-14597-X**[7]**Elliott Stein,*Free actions on products of spheres*, Michigan Math. J.**26**(1979), no. 2, 187–193. MR**532319****[8]**L. Dornhoff,*Group representation theory*, Dekker, New York, 1971-1972.

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1981-0603774-2

Article copyright:
© Copyright 1981
American Mathematical Society