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.