I know that $S_3$ is the semidirect product of $\bigl\langle(1\ \ 2\ \ 3)\bigr\rangle \rtimes\bigl\langle(1\ \ 2)\bigr\rangle$, and I'm not sure where exactly the direct product property fails. Is it only because $\bigl\langle(1\ \ 2)\bigr\rangle$ is not normal in $S_3$?
2026-02-22 18:56:15.1771786575
Why is the symmetry group $S_3$ not the direct product of two nontrivial groups?
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That is enough, yes. If $G=H_1\times H_2$, then $\{e_{H_1}\}\times H_2$ and $H_1\times\{e_{H_2}\}$ are normal subgroups of $G$.
Besides, the direct product of two abelian groups is again abelian.