S3 Is the Smallest Nonabelian Semidirect Product

The symmetric group on three letters is C_3 semidirect C_2 with inversion as the nontrivial action. I want the notation, the mechanism, and the failure mode visible at the same time.

Notation

A semidirect product combines groups $N$ and $H$ after choosing an action $\alpha:H\to\operatorname{Aut}(N)$. A presentation records generators and relations but may conceal the size of the group.

$$ S_3=\langle r,s\mid r^3=s^2=e,\ srs=r^{-1}\rangle $$

I keep the defining relation $\mathsf D$ above the derived relation $\mathsf C$. This exposes whether cancellation used $x\ne0$ and whether the conclusion is canonical.

$$ S_3\cong C_3\rtimes C_2 $$

Stress the formula

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ \left\{\begin{aligned}r&=(1\,2\,3),\\s&=(1\,2),\\srs&=(1\,3\,2)=r^{-1}\end{aligned}\right. $$

Interpretation

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad S_3=\langle r,s\mid r^3=s^2=e,\ srs=r^{-1}\rangle,\\[5pt] \mathsf{C}\;&:\quad S_3\cong C_3\rtimes C_2. \end{aligned} $$

Limit of the argument

Changing the action $\alpha$ can change the group even when $N$ and $H$ remain fixed. The direct product is only the special case where the action is trivial.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] S_3\cong C_3\rtimes C_2 \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Fri 17 May 2019. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.

S3 Is the Smallest Nonabelian Semidirect Product by Vicfred

Notation

Stress the formula

Interpretation

Limit of the argument