Stabiliser is a subgroup

by Patrick Stevens Jun 20 2016 updated Jul 7 2016

Given a group acting on a set, each element of the set induces a subgroup of the group.

[summary: Given a group $~$G$~$ acting on a set $~$X$~$, the stabiliser of some element $~$x \in X$~$ is a subgroup of $~$G$~$. ]

Let $~$G$~$ be a Group which acts on the set $~$X$~$. Then for every $~$x \in X$~$, the stabiliser $~$\mathrm{Stab}_G(x)$~$ is a subgroup of $~$G$~$.


We must check the group axioms.