# Free actions of finite groups on products of even-dimensional spheres

Suppose a finite 2-group G acts freely on X = $\prod_{i=1}^k$ *S*$^{2n_i}$, a product of k even-dimensional spheres, k > 2. Is it possible for G to be non-abelian? What if we additionally assume that spheres in the product are equidimensional?

Some comments: The equality 2k = $\chi(X)$ = |G|$\chi(X/G)$ coming from the covering X $\to$ X/G ensures that a finite group acting freely on X is a 2-group and also answers the question for k = 1, 2. (Actually, for k = 1 this gives a proof of a classical theorem: the only group which can act freely on an even-dimensional sphere is the cyclic group of order 2. Does anyone know who is this result originally due to? Sorry for a question inside the question; perhaps someone can comment on this one.) Hence if one would like to construct a sort of "minimal" example, it should involve an action of either the quaternion group Q8 or the dihedral group Dih4 for k = 3. I thought about this a little bit, but I feel like I'm not comfortable enough with non-abelian groups.

I've stumbled across some papers where authors characterize arbitrary finite groups acting freely on X in terms of existence of particular representations, but explicit examples are given only in the abelian case.

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[Adem, Alejandro; Davis, James F. Topics in transformation groups. Handbook of geometric topology, 1--54, North-Holland, Amsterdam, 2002. MR1886667] mentions the fact that a finite $2$-group such that every element of order $2$ is central acts freely on $(S^{|G|/2-1})^k$, where $k$ is the rank of $G$, that is, the rank of the biggest subgroup of the form $(\mathbb Z_2)^{(r)}$: «The action is built by inducing up sign representations on $k$ elements of order $2$ which span the unique central elementary abelian subgroup of $G$ and then taking their product».
Using GAP I find a smallest non abelian example $G$ of order 32, which it describes as $(C_4\times C_2):C_4$.
I guess $C_2\times C_2\times Q_8$ is a simpler example... – Mariano Suárez-Alvarez Mar 29 '10 at 23:45