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

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 $p|(n + 1)$. 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 ${2^{\mu (p)}}|(n + 1)$, where $\mu (p)$ is an integer depending on $p$.