I have been reading through some book, such as Geometry of Crystallographic groups (by Andrzej Szczepanski), and during this reading I came across the relationship: E(n) = O(n) ⋉ $\mathbb{R}^n$. Unfortunately there was no proof attached. I was hoping someone could help me with the proof for this? I'm not really sure how to compare the pairs of elements in O(n) ⋉ $\mathbb{R}^n$ to the Isometries in E(n). Any help would be greatly appreciated.
2026-03-28 09:54:24.1774691664
Help understanding/proving : E(n) = O(n) ⋉ $\mathbb{R}^n$
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