| course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
15 |
IB |
2020 |
Groups, Rings and Modules |
|
Paper 2, Section I, G |
2020 |
Assume a group $G$ acts transitively on a set $\Omega$ and that the size of $\Omega$ is a prime number. Let $H$ be a normal subgroup of $G$ that acts non-trivially on $\Omega$.
Show that any two $H$-orbits of $\Omega$ have the same size. Deduce that the action of $H$ on $\Omega$ is transitive.
Let $\alpha \in \Omega$ and let $G_{\alpha}$ denote the stabiliser of $\alpha$ in $G$. Show that if $H \cap G_{\alpha}$ is trivial, then there is a bijection $\theta: H \rightarrow \Omega$ under which the action of $G_{\alpha}$ on $H$ by conjugation corresponds to the action of $G_{\alpha}$ on $\Omega$.