A note on the normalizer of Sylow ۲-subgroup of special linear‎ ‎group {\rm SL}_۲(p^f)

Let G={\rm SL}_۲(p^f) be a special linear group and P be a Sylow‎ ‎۲-subgroup of G‎, ‎where p is a prime and f is a positive‎ ‎integer such that p^f>۳‎. ‎By N_G(P) we denote the normalizer of‎ ‎P in G‎. ‎In this paper‎, ‎we show that N_G(P) is nilpotent (or‎ ‎۲-nilpotent‎, ‎or supersolvable) if and only if p^{۲f}\equiv‎ ‎۱\,({\rm mod}\,۱۶)‎.

کلیدواژه ها:

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