2019-30.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	30	IB 2019 Groups, Rings and Modules	Paper 4, Section I, G	2019

Let $G$ be a group and $P$ a subgroup.

(a) Define the normaliser $N_{G}(P)$.

(b) Suppose that $K \triangleleft G$ and $P$ is a Sylow $p$-subgroup of $K$. Using Sylow's second theorem, prove that $G=N_{G}(P) K$.