2017-32.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	32	IB 2017 Groups, Rings and Modules	Paper 1, Section II, 10E	2017

(a) State Sylow's theorem.

(b) Let $G$ be a finite simple non-abelian group. Let $p$ be a prime number. Show that if $p$ divides $|G|$, then $|G|$ divides $n_{p} ! / 2$ where $n_{p}$ is the number of Sylow $p$-subgroups of $G$.

(c) Let $G$ be a group of order 48 . Show that $G$ is not simple. Find an example of $G$ which has no normal Sylow 2-subgroup.