Construction of one-fixed-point actions on spheres of nonsolvable groups I

Abstract

Let $G$ be a finite group. It is known that if a homotopy sphere $X$ has a one-fixed-point smooth $G$-action then the dimension of $X$ is greater than or equal to $6$. It is also known that there is an effective $2$-pseudofree one-fixed-point smooth $G$-action on the sphere $S^n$ of dimension $n$ if and only if $n$ is equal to $6$ and $G$ is isomorphic to the alternating group $A_5$ on five letters. E. Stein proved that for the group $G = {\textrm{SL}}(2, 5) \times Z_m$ such that $m$ is prime to $30$, there is a $3$-pseudofree one-fixed-point smooth $G$-action on $S^7$, where $Z_m$ is a cyclic group of order $m$. In this article, we determine the finite groups $G$ possessing $3$-pseudofree one-fixed-point smooth $G$-actions on $S^6$. In addition, for an arbitrary finite group $G$ isomorphic to $A_5$, $A_5 \times Z_2$, or ${\textrm{SL}}(2, 5) \times Z_m$ such that $m$ is prime to $30$, we prove that there is a $3$-pseudofree one-fixed-point smooth $G$-action on $S^7$.