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

Document Type : Research Paper

Author

Yantai University

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 $p^{2f}\equiv‎ ‎1\,({\rm mod}\,16)$‎.

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#### References

B. Huppert (1967). Endliche Gruppen I. Springer-Verlag, Berlin. D. J. S. Robinson (1996). A Course in the Theory of Groups. (Second Edition), Springer-Verlag, New York.