Group acts on left coset space of subgroup by left multiplication

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 $$G$$.