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Let $G$ be a group and $H$ its subgroup. Let $n=[G:H]$ be a cardinal number. Let $C=\{aH\,|\,a\in G\}.$ We have $n=\operatorname{card}(C).$ We define for any $g\in G$ the map $\phi_g:C\to C$ by the formula $\phi_g(aH)=gaH.$ The maps $\phi_g$ are well-defined because if $aH=bH,$ then $gaH=gbH$ for any $g\in G.$

$\phi_g$ must be 1-1 because if $gaH=gbH,$ then $aH=g^{-1}(gaH)=g^{-1}(gbH)=bH.$ $\phi_g$ must be onto because for $a,g\in G$ we have $aH=g(g^{-1}aH)=\phi_g((g^{-1}a)H).$

Therefore, $\phi_g$ is a permutation of $C.$ We can say that $\{\phi_g\,|\,g\in G\}\subseteq \operatorname{Sym}(n).$

Let $f:G\to\operatorname{Sym}(n)$ be defined by the formula

$f(g)=\phi_g.$

$f$ is a homomorphism because for $g_1,g_2\in G$ and $aH\in C$ we have

$(f(g_1g_2))(aH)=\phi_{g_1g_2}(aH)=g_1g_2aH=(\phi_{g_1}\circ\phi_{g_2})(aH)=(f(g_1)\circ f(g_1))(aH).$

I've just noticed this. Is there a name for this homomorphism? Are there names for its kernel and image?

Edit: Let's rename $f$ to $f_l$ and define $f_r$ analogously, but with right cosets instead of left cosets. Can it be for some $G$ and $H$ that

$\ker f_l\neq \ker f_r?$

Edit: OK, I think it can't be. We have

$ \begin{eqnarray} \ker f_l&=&\{g\in G \,|\, \phi_g=\operatorname{id}\}\\&=&\{g\in G\,|\,(\forall a\in G) gaH=aH\}\\&=&\{g\in G\,|\,(\forall a\in G) a^{-1}gaH=H\}\\&=&\{g\in G\,|\,(\forall a\in G) a^{-1}ga\in H\} \end{eqnarray} $

Analogously,

$ \ker f_r = \{g\in G\,|\,(\forall a\in G) aga^{-1}\in H\} $

and $ (\forall a\in G) a^{-1}ga\in H\iff (\forall a\in G) aga^{-1}\in H $

So the kernels are equal.

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    It's somewhat more accurate to describe this as a map from G to $\operatorname{Sym}(G/H)$, the group of permutations of the coset space $G/H = \{aH:a\in G\}$. This latter group is *isomorphic* to $\operatorname{Sym([G:H])}$, but there isn't a canonical isomorphism between these groups -- it depends on choosing an ordering of $G/H$.2012-02-11

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$f$ is called a coset representation. The left and right coset reprsentations are isomorphic via the bijection sending $gH$ to $Hg^{-1}$.

Since the actions are isomorphic they have the same kernel. This kernel is called the normal core.