Question

If f is an automorphism of ${\mathit{S}}_{\mathbf{3}}$, prove that there exists $\mathit{\sigma }\mathbf{\in }{\mathit{S}}_{\mathbf{3}}$such that $\mathit{f}\left(\tau \right)\mathbf{=}\mathit{\sigma }\mathit{\tau }{\mathit{\sigma }}^{\mathbf{-}\mathbf{1}}$for every $\mathit{\tau }\mathbf{\in }{\mathit{S}}_{\mathbf{3}}$.

Short answer

It is proved that there exists $\sigma \in S_3$such that $f\left(\tau \right)=\sigma \tau {\sigma }^{-1}$for every $\tau \in S_3$.

Step-by-step solution

Given

We know that $S_3$is generated by $\left(12\right)$and $\left(123\right)$and we know that automorphism f of$S_3$ must in particular preserve the order of the elements, so$f\left(\left(12\right)\right)\in \left\{\left(12\right),\left(13\right),\left(23\right)\right\}$ and$f\left(\left(123\right)\right)\in \left\{\left(123\right),\left(132\right)\right\}$ .Therefore, we have atmost 6 automorphism of $f\left(\left(123\right)\right)\in \left\{\left(123\right),\left(132\right)\right\}$.

Form the table and prove the result

Form the table that gives $\sigma$such that $f\left(\tau \right)=\sigma \tau {\sigma }^{-1}$for each image of $\left(12\right)$and $\left(123\right)$by f as:

$f\left(\left(12\right)\right)$	$f\left(\left(123\right)\right)$	$\sigma$
$\left(12\right)$	$\left(123\right)$	$\left(1\right)$
$\left(13\right)$	$\left(123\right)$	$\left(132\right)$
$\left(23\right)$	$\left(123\right)$	$\left(123\right)$
$\left(12\right)$	$\left(132\right)$	$\left(12\right)$
$\left(13\right)$	$\left(132\right)$	$\left(23\right)$
$\left(23\right)$	$\left(132\right)$	$\left(13\right)$

Hence, it is proved that there exists $\sigma \in S_3$such that $f\left(\tau \right)=\sigma \tau {\sigma }^{-1}$for every $\tau \in S_3$.