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Transactions of the American Mathematical Society
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
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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$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0603774-2
PII: S 0002-9947(1981)0603774-2
Article copyright: © Copyright 1981 American Mathematical Society