Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some restrictions on finite groups acting freely on $ (S\sp{n})\sp{k}$

Author: Gunnar Carlsson
Journal: Trans. Amer. Math. Soc. 264 (1981), 449-457
MSC: Primary 55M35; Secondary 55S10
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 $ {({S^n})^k}$, with trivial action on homology. It is shown that elements $ x$ of order $ p$, $ p$ an odd prime, with $ x$ in the normalizer of an elementary abelian $ 2$-subgroup $ E$ of $ G$, must act trivially on $ E$ unless $ p\vert(n + 1)$. It is also shown that if $ p = 3$ or $ 7$, $ x$ must act trivially, independent of $ n$. This produces a large family of groups which do not act freely on $ {({S^n})^k}$ for any values of $ n$ and $ k$. For certain primes $ p$, using the mod two Steenrod algebra, one may show that $ x$ acts trivially unless $ {2^{\mu (p)}}\vert(n + 1)$, where $ \mu (p)$ is an integer depending on $ p$.

References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams and C. W. Wilkerson, Finite $ H$-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 $ {S^n} \times {S^n}$, Ann. of Math. (2) 66 (1957), 586-588. MR 0096235 (20:2725)
  • [4] J. Milnor, Groups which act on $ {S^n}$ 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.

Similar Articles

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

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

Additional Information

Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society