Why is the identity map never equal to the product of an odd number of reflections?

Question

Suppose I have an some plane and an identity mapping on the points of the plane. I see that the identity can be expressed as a product of an even number of reflections, since any reflection has itself as its own inverse. But why is it impossible to ever express the identity as the product of an odd number of reflections? Thanks.

Answer 1

On lhf's request I make my comment into an answer:

A reflection reverses the orientation of the plane: If you look what happens to a triangle whose corners are $ABC$ (oriented counterclockwise) then after a reflection the corners will be ordered as $ACB$ (oriented counterclockwise). So after applying an odd number of reflections the corners of $ABC$ will again be arranged as $ACB$ counterclockwise, so you can't possibly have the identity.

If you want to make this into a rigorous proof, you can do this by introducing the determinant (of the linear part) of an isometry and show that it is multiplicative: That is to say $\det(ST) = \det(S)\det(T)$. Now a reflection has determinant $-1$, so the determinant of the composition of $n$ reflections will be $(-1)^{n}$. If $n$ is odd then the determinant will be $(-1)^n = (-1)^{2k+1} = -1$. On the other hand the determinant of the identity transformation is $1$, so only an even number of reflections can give the identity.