A note on the normalizer of Sylow $2$-subgroup of special linear‎ ‎group ${rm SL}_2(p^f)$

Abstract

Let $G={rm SL}_2(p^f)$ be a special linear group and $P$ be a Sylow‎ ‎$2$-subgroup of $G$‎, ‎where $p$ is a prime and $f$ is a positive‎ ‎integer such that $p^f>3$‎. ‎By $N_G(P)$ we denote the normalizer of‎ ‎$P$ in $G$‎. ‎In this paper‎, ‎we show that $N_G(P)$ is nilpotent (or‎ ‎$2$-nilpotent‎, ‎or supersolvable) if and only if $p^{2f}equiv‎ ‎1,({rm mod},16)$‎.