Permutation of Cosets

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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:

$(1): \quad$ For any $g \in G$, the mapping $\theta_g: \mathbb S \to \mathbb S$ defined by $\map {\theta_g} {x H} = g x H$ is a permutation of $\mathbb S$.
$(2): \quad$ The mapping $\theta$ defined by $\map \theta g = \theta_g$ is a homomorphism from $G$ into the symmetric group on $\mathbb S$.
$(3): \quad$ The kernel of $\theta$ is the subgroup $\ds \bigcap_{x \mathop \in G} x H x^{-1}$.


Corollary 1

Let $G$ be a group.

Let $H \le G$ such that $\index G H = n$ where $n \in \Z$.


Then:

$\exists N \lhd G: N \lhd H: n \divides \index G H \divides n!$


Corollary 2

Let $G$ be a group.

Let $p$ be the smallest prime such that:

$p \divides \order G$

where $\divides$ denotes divisibility.


Let $\exists H: H \le G$ such that $\order H = p$.

Then $H$ is a normal subgroup of $G$.


Proof

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

\(\ds x H\) \(=\) \(\ds y H\)
\(\ds \leadstoandfrom \ \ \) \(\ds y^{-1} x\) \(\in\) \(\ds H\)
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {g y}^{-1} g x\) \(=\) \(\ds y^{-1} x \in H\)
\(\ds \leadstoandfrom \ \ \) \(\ds \map {\theta_g} {y H}\) \(=\) \(\ds \map {\theta_g} {x H}\)


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


Then we see that $\forall x H \in \mathbb S: \map {\theta_g} {g^{-1} x H} = x H$, 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:

\(\ds \map {\theta_{u v} } {x H}\) \(=\) \(\ds u v x H\)
\(\ds \) \(=\) \(\ds \map {\theta_u} {v x H}\)
\(\ds \) \(=\) \(\ds \map {\theta_u} {\map {\theta_v }{x H} }=\)


This shows that $\theta_{u v} = \theta_u \theta_v$, and thus:

$\map \theta {u v} = \map \theta u \, \map \theta v$

Thus $\theta$ is a homomorphism.


Now to calculate $\map \ker \theta$:

\(\ds \map \ker \theta\) \(=\) \(\ds \set {g \in G: \theta_g = I_\mathbb S}\)
\(\ds \) \(=\) \(\ds \set {g \in G: \forall x \in G: \map {\theta_g} {x H} = x H}\)
\(\ds \) \(=\) \(\ds \set {g \in G: \forall x \in G: g x h = x H}\)
\(\ds \) \(=\) \(\ds \set {g \in G: \forall x \in G: x^{-1} g x h = H}\)
\(\ds \) \(=\) \(\ds \set {g \in G: \forall x \in G: x^{-1} g x \in H}\)
\(\ds \) \(=\) \(\ds \set {g \in G: \forall x \in G: g \in x H x^{-1} }\)
\(\ds \) \(=\) \(\ds \bigcap_{x \mathop \in G} x H x^{-1}\)

as required.

$\blacksquare$


Sources

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