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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?

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1 Answer 1

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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.

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