Math Is Fun Forum

alexGo

Guest

Another Groups Question

Let N be a normal subgroup of the orthogonal group O(2). Show that if N contains a reflection in some line through the origin, then N = O(2).

I'm not really sure where to go with this. I know that, because N is a normal subgroup, ANA-¹ = N for all A ∈ O(2) (where N is a set). I know that AA^T = I for all A ∈ O(2), and I know that if R is a reflection, then R = R-¹. I can't seem to pull these together into a proof though.

Am I correct in saying that all elements in O(2) are either reflections or rotations? I know that two reflections generate a rotation...

Any hints?

Thanks

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Another Groups Question

I know that, because N is a normal subgroup, ANA-¹ = N for all A ∈ O(2)

What does this mean, with respect to the reflection that is in N?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

alexGo

Guest

Re: Another Groups Question

I'm not sure. It means that ARA-¹ = some element in N for all A ∈ O(2)...

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Another Groups Question

One important fact about the Orthogonal group is that it's generated by reflections.  Assuming you're ok with this, your goal is to prove that all reflections are in N.  Use your previous post to do this.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."