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

Chapter 14: Symmetries defines the orthogonal group as follows:

O(n, ) = {A GL(n, )| = } where GL(n, ) 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 has det A =1 or exactly half do"

Would appreciate some help with this exercise.

Peter