# How many “elementary” isometries are needed to generate any isometry of an n-dimensional Euclidean space?

I use here the term Euclidean space in the rigorous sense of an affine space over $\mathbb{R}^n$, equipped with the Euclidean inner product (see here). Let $\mathbb{E}^n$ be the Euclidean $n-$space.

Consider the set $S$ of all isometries on $\mathbb{E}^n$. Given any element $f\in S$, intuitively I would expect $f$ to be equivalent to a composition $g$ of the three "elementary" isometries, i.e., rotations, translation and reflections. With equivalent, I mean that $f(P)=g(P) \ \forall P \in \mathbb{E}^n$.

• Is this true? Since "elementary" isometries are affine functions (they can all be represented by a multiplication by a matrix or by addition of a constant, i.e., a translation), and since the composition of two affine functions is affine (I think), this would imply that all isometries on $\mathbb{E}^n$ are affine functions, right?
• If it is, how many "elementary" isometries are needed to generate any isometry? For example, can we say that for $n\ge2$, any isometry is a composition of a rotation $or$ a reflection plus a translation?

In a Euclidean vector space of dimension $n$, any isometry is the product of at most $n$ reflections.
The proof consists in defining by induction $r_1, \dots, r_n$ reflections such that $r_1 \circ f$ stabilizes a line, $r_2 \circ r_1 \circ f$ stabilizes à plane and so on where $f$ is the isometry that you are looking to decompose as a product of reflections.
Knowing that a translation is a product of $2$ reflections, you can also conclude that any affine isometry in an affine space of dimension $n$ is the product of affine reflections,at most $n+2$.