Every finite group of congruences of the $n$-dimensional Euclidean space has a fixed point

Question

Is there an "absolute" proof of the fact that if $G$ is a finite group of congruences of the $n$-dimensional Euclidean space, then $G$ has a fixed point?

Answer 1

I am assuming a "congruence" of $\mathbb{R}^n$ is an affine transformation, e.g. a map of the form $v \mapsto Tv + a$ where $T \in M_n (\mathbb{R})$ and $a \in \mathbb{R}^n$. Let's denote by $\text{Aff}(\mathbb{R}^n)$ the group of all affine transformations of $\mathbb{R}^n$, and suppose $G \le \text{Aff}(\mathbb{R}^n)$ is a finite subgroup.

Consider $x:= \frac{1}{|G|} \sum_{f \in G} f(0)$. For any $g \in G$, the map $y \mapsto g(y) - g(0)$ is linear; hence, we may compute $$g(x) = g(x) - g(0) + g(0) = \frac{1}{|G|} \sum_{f \in G} (g(f(0)) - g(0)) + g(0) = \frac{1}{|G|} \sum_{f \in G} g(f(0)) = x.$$ Thus, $x$ is a fixed point of $G$.