Originally Posted by

**Bernhard** I am reading Aigli Papantonopoulou's book Algebra: Pure and applied.

Chapter 14: Symmetries defines the orthogonal group as follows:

O(n, $\displaystyle \mathbb{R}$) = {A $\displaystyle \in$GL(n,$\displaystyle \mathbb{R}$)| $\displaystyle A^T$ = $\displaystyle A^{-1}$} where GL(n,$\displaystyle \mathbb{R}$) is the general linear group.

I need some help with the following problem: Problem 12 of Exercises 14

"Let G be a subgroup of O(n). Show that either every element A$\displaystyle \in$ has det A =1 or exactly half do"

Would appreciate some help with this exercise.

Peter