a product of $f+1$ spheres with trivial action on homology.

We prove this theorem using a recursive method for constructing free actions. This method involves the construction of a $G$-equivariant spherical fibration $p : E \to X$ over a given finite $G$-CW-complex $X$ in each step. We require that the total space $E$ is also homotopy equivalent to a finite $G$-CW-complex and that the $G$-action on $E$ has smaller isotropy than the $G$-action on $X$. Once they are constructed, by taking fiber joins these fibrations can be replaced by $G$-fibrations which are non-equivariantly homotopy equivalent to trivial fibrations. This gives a $G$-action on a finite CW-complex $Y$ homotopy equivalent to $X \times {\mathbb S}^N$ for some $N>0$. Using this method, after some steps, one gets a free action on a product of spheres. This method was first developed by Connolly and Prassidis [@connolly-prassidis] and later used also in [@adem-smith] and [@unlu-thesis].

Since every finite group has a faithful complex representation, every finite group has a complex representation with fixity $f$ for some positive integer $f$. Hence, as a corollary of Theorem \[thm:main\] we obtain an affirmative answer to Question \[ques:jdavis\] in homotopy category.

\[cor:main\] Every finite group acts freely and 