Is this group finite?

Question

Let $G$ be a sub-group of the invertible real matrices of size $n$ (usually noted $GL_n(\mathbb{R})$), such that $\forall M\in G,M^2=I_n$

Is $G$ finite ?

Answer 1

Yes. The group will be finite. It was given that all the elements of the group $G$ satisfy the equation $M^2=I_n=1_G$. A standard exercise is to show that this implies that $G$ is abelian. All the elements of $G$ are diagonalizable (finite order, eigenvalues $\pm1$ only), so another standard exercise shows that they are simultaneously diagonalizable.

So after conjugation we can assume that $G$ consists of diagonal matrices with all the entries $\pm1$. Therefore $|G|\le 2^n$.