A note on the normalizer of Sylow 2-subgroup of special linear group $SL_2(p^f)$

Abstract

Let $G=SL_2(p^f)$ 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 $p^{2f}equiv 1,(mod 16)$.

Keywords

• special linear group
• Sylow subgroup
• normalizer
• nilpotent
• supersolvable