I am looking for the alternative construction of the twisted finite simple group $^2 E_6 (q)$ possibly avoiding the Lie theory. Especially I am interested in calculating its order. Maybe some of you ...
$G$ be a finite group and $f \in Aut (G)$ such that $f^3$ is identity and $f$ has unique fixed point , then any $p$-Sylow subgroup is normal?
Let $G$ be a finite group and $f \in Aut (G)$ such that $f^3$ is identity and $f(x)=x \implies x=e$ ; then is it true that for every prime $p$ dividing $|G|$ , there is exactly one $p$-Sylow subgroup ...