- #1

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- TL;DR Summary
- Using the orbit-stabilizer theorem to identify groups.

I want to identify:

The orbit-stabilizer theorem would give us the result, but my problem is to apply it. My problem is how to find the stabilizer.

In 1 how to define the action of ##O(n+1,R)## on ##S^n## and then conclude that ##stab(x)=O(n,R)## for ##x \in S^n##. And similar in 2?

It would be helpful and appreciated if you simplify/explain the method for showing this.

- ##S^n## with the quotient of ##O(n + 1,R)## by ##O(n,R)##.
- ##S^{2n+1}## with the quotient of ##U(n + 1)## by ##U(n)##.

The orbit-stabilizer theorem would give us the result, but my problem is to apply it. My problem is how to find the stabilizer.

In 1 how to define the action of ##O(n+1,R)## on ##S^n## and then conclude that ##stab(x)=O(n,R)## for ##x \in S^n##. And similar in 2?

It would be helpful and appreciated if you simplify/explain the method for showing this.