Show that SO(n) is a normal subgroup of O(n).
A normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.
SO(n) is the set of orthogonal matrices of determinant 1.
O(n) is the set of real matrices whose inverses equal their transposes (orthogonal matrices).
I'm simply bad at writing proofs. Please help?
Best Answer
Simply note that $SO(n)$ is the kernel of the group homomorphism $$ \det :O(n) \longrightarrow \mathbb{R}^* \ \ \ \ \ A \mapsto \det A$$
Recall that every kernel is a normal subgroup.