Group acts on left coset space of subgroup by left multiplication

This article describes a pivotal group action: a group action on a set closely associated with the group. This action is important to understand and remember.
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Statement

Suppose $G$ is a group and $H$ is a subgroup of $G$. Consider the left coset space $G/H$, i.e., the collection of left cosets of $H$ in $G$:

$G/H:=\{xH\mid x\in G\}$.

Then there exists a natural group action of $G$ on $G/H$ by left multiplication:

$g.(xH)=(gx)H$.

In other words, left multiplying any left coset by an element of the group yields a left coset, and this defines an action of the group on the left coset space by left multiplication. Further, the following facts are true about this group action:

1. The stabilizer of the coset $xH$ is the conjugate subgroup $xHx^{-1}$.
2. The intersection of all stabilizers, which is also the kernel of the homomorphism $G\to \operatorname {Sym} (G/H)$ associated with this action, is the intersection of all conjugate subgroups of $H$ in $G$. This is termed the normal core of $H$ in $G$.