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

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

MathSciNet review:
603774

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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**-spaces and algebras over the Steenrod algebra*, Ann. of Math.**111**(1980), 95-143. MR**558398 (81h:55006)****[2]**G. Carlsson,*On the non-existence of free actions of elementary abelian groups on products of spheres*, Amer. J. Math.**102**(1980), 1147-1157. MR**595008 (82a:57038)****[3]**P. Conner,*On the action of a finite group on*, Ann. of Math. (2)**66**(1957), 586-588. MR**0096235 (20:2725)****[4]**J. Milnor,*Groups which act on**without fixed points*, Amer. J. Math.**79**(1957), 623-630. MR**0090056 (19:761d)****[5]**R. Oliver, Proc. Aarhus Sympos. on Algebraic Topology, 1978, Lecture Notes in Math., vol. 763, Springer-Verlag, Berlin and New York.**[6]**R. P. Stanley,*Invariants of finite groups and their applications to combinatorics*, Bull. Amer. Math. Soc. (N.S.)**1**(1979), 475-511. MR**526968 (81a:20015)****[7]**E. Stein,*Free actions on products of spheres*, Michigan Math. J. (to appear). MR**532319 (80j:57042)****[8]**L. Dornhoff,*Group representation theory*, Dekker, New York, 1971-1972.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
55M35,
55S10

Retrieve articles in all journals with MSC: 55M35, 55S10

Additional Information

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

Article copyright:
© Copyright 1981
American Mathematical Society