# Affine group and semidirect product

I proved that $$\mathrm{Aff}(n) \cong O(n) \rtimes \mathbb{R}^n$$ and $$\mathrm{Aff}(n) \cong \mathbb{R}^n \rtimes O(n)$$, where $$\mathrm{Aff}(n)$$ is the affine group, $$O(n)$$ the orthogonal group, $$\rtimes$$ denotes the semi-direct product of two groups.

Does it make sense that $$\mathrm{Aff}(n) \cong \mathbb{R}^n \rtimes O(n)$$ holds? I saw that most sources tend write down $$\mathrm{Aff}(n) \cong O(n)\rtimes \mathbb{R}^n$$ and as a consequence I am not sure whether my proof of showing $$\mathrm{Aff}(n) \cong \mathbb{R}^n \rtimes O(n)$$ makes sense. I could give the proof if there is interest in that.

• Where did you see $O(n)\rtimes \mathbb{R}^n$? Are you sure it wasn't $O(n)\ltimes \mathbb{R}^n$? – FredH Mar 31 at 15:29
• What is the difference between those expressions? – Dani Mar 31 at 16:25
• With $\rtimes$, the normal subgroup is on the left; with $\ltimes$, it is on the right. – FredH Mar 31 at 16:26
• I think it was $\ltimes$. But then why is $O(n) \rtimes \mathbb{R}^n \cong \mathrm{Aff}(n)$ not true? Both $O(n)$ and $\mathbb{R}^n$ are normal subgroups right? – Dani Mar 31 at 16:30
• Because $O(n)$ is not a normal subgroup of $\mathrm{Aff}(n)$. – FredH Mar 31 at 16:33