Let $G$ be a group, let $H \le G$, and let $G/H := \{gH : g \in G\}$ denote the set of left cosets of $H$ in $G$. Let $\operatorname{Sym}(G/H)$ denote the group of all bijections $G/H \to G/H$ under composition, and let $\operatorname{id}_{G/H}: G/H \to G/H$ denote the identity permutation. Define the left coset action homomorphism $\lambda: G \to \operatorname{Sym}(G/H)$ by $\lambda(x)(gH) = (xg)H$ for all $x,g \in G$. Then

\begin{align*} \ker \lambda = \operatorname{Core}_G(H). \end{align*}

Here

\begin{align*} \operatorname{Core}_G(H) := \bigcap_{g \in G} gHg^{-1}. \end{align*}