Permutation of Cosets

Theorem
Let $$G$$ be a group and let $$H \le G$$.

Let $$\mathbb{S}$$ be the set of all distinct left cosets of $$H$$ in $$G$$.

Then:
 * 1) For any $$g \in G$$, the mapping $$\theta_g: \mathbb{S} \to \mathbb{S}$$ defined by $$\theta_g \left({x H}\right) = g x H$$ is a permutation of $$\mathbb{S}$$.
 * 2) The mapping $$\theta$$ defined by$$\theta \left({g}\right) = \theta_g$$ is a homomorphism from $$G$$ into the Symmetric Group on $$\mathbb{S}$$.
 * 3) The kernel of $$\theta$$ is the subgroup $$\bigcap_{x \in G} x H x^{-1}$$.

Corollary
Let $$H \le G$$ such that $$\left[{G : H}\right] = n$$ where $$n \in \mathbb{Z}$$.

Then $$\exists N \triangleleft G: N \triangleleft H: n \backslash \left[{G : N}\right] \backslash n!$$.

Proof

 * First we need to show that $$\theta_g$$ is well-defined, and injective.

$$ $$ $$ $$

Thus $$\theta_g$$ is well-defined, and injective.

Then we see that $$\forall x H \in \mathbb{S}: \theta_g \left({g^{-1} x H}\right) = x H$$, so $$\theta_g$$ is surjective.

Thus $$\theta_g$$ is a well-defined bijection on $$\mathbb{S}$$, and therefore a permutation on $$\mathbb{S}$$.


 * Next we see:

$$ $$ $$

This shows that $$\theta_{u v} = \theta_u \theta_v$$, and thus $$\theta \left({u v}\right) = \theta \left({u}\right) \theta \left({v}\right)$$.

Thus $$\theta$$ is a homomorphism.


 * Now to calculate $$\mathrm{ker} \left({\theta}\right)$$.

$$ $$ $$ $$ $$ $$ $$

as required.

Proof of Corollary
Apply the main result to $$H$$ and let $$N = \mathrm{ker} \left({\theta}\right)$$.

Then $$N \triangleleft G$$ and $$N \triangleleft H$$ so $$H / N \le G / N$$ such that $$\left[{G / N : H / N}\right] = n$$ from the Correspondence Theorem.

Thus $$n \backslash \left[{G : N}\right]$$.

Also by the main result, $$\exists K \in S_n: G / N \cong K$$.

Thus $$\left[{G : N}\right] \backslash n!$$ as required.