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 $S^7$.
Funding Statement
This research was partially supported by JSPS KAKENHI Grant Number 18K03278
Acknowledgments
The referee read carefully the manuscript and contributed to brushing up the manuscript. The author would like to express his gratitude to the referee for it.
Citation
Masaharu Morimoto. "Construction of one-fixed-point actions on spheres of nonsolvable groups I." Osaka J. Math. 60 (3) 493 - 525, July 2023.
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