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2

$begingroup$

We will say that $G<GL(n;mathbbR)$ is an orthogonal group if $G$ is abstractly isomorphic to $O(p,q)$ for some $p+q=n$.
If $Gcong O(p,q)$ is an orthogonal group, is $G$ actually conjugate in $GL(n;mathbbR)$ to $O(p,q)$?

I think this follows for groups $G$ abstractly isomorphic to $O(n)$ as this is the maximal compact subgroup and hence unique up to conjugacy. I'm looking for suggestions on how to go about proving (or looking for counterexamples) to the indefinite cases.

linear-algebra lie-groups

share|cite|improve this question

edited Mar 15 at 16:17

Steve

asked Mar 15 at 5:26

SteveSteve

61949

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The wording here is a bit circular. Do you mean, If two subgroups of $GL(n, Bbb R)$ that are both isomorphic to $O(p, q)$ for some $p, q$ (such that $p + q = n$), are they necessarily conjugate?
  $endgroup$
  – Travis
  Mar 15 at 8:35
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, that's a good way to say it!
  $endgroup$
  – Steve
  Mar 15 at 16:18
  

  
  
  
  

add a comment | 

2

$begingroup$

We will say that $G<GL(n;mathbbR)$ is an orthogonal group if $G$ is abstractly isomorphic to $O(p,q)$ for some $p+q=n$.
If $Gcong O(p,q)$ is an orthogonal group, is $G$ actually conjugate in $GL(n;mathbbR)$ to $O(p,q)$?

I think this follows for groups $G$ abstractly isomorphic to $O(n)$ as this is the maximal compact subgroup and hence unique up to conjugacy. I'm looking for suggestions on how to go about proving (or looking for counterexamples) to the indefinite cases.

linear-algebra lie-groups

share|cite|improve this question

edited Mar 15 at 16:17

Steve

asked Mar 15 at 5:26

SteveSteve

61949

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The wording here is a bit circular. Do you mean, If two subgroups of $GL(n, Bbb R)$ that are both isomorphic to $O(p, q)$ for some $p, q$ (such that $p + q = n$), are they necessarily conjugate?
  $endgroup$
  – Travis
  Mar 15 at 8:35
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, that's a good way to say it!
  $endgroup$
  – Steve
  Mar 15 at 16:18
  

  
  
  
  

add a comment | 

$begingroup$

We will say that $G<GL(n;mathbbR)$ is an orthogonal group if $G$ is abstractly isomorphic to $O(p,q)$ for some $p+q=n$.
If $Gcong O(p,q)$ is an orthogonal group, is $G$ actually conjugate in $GL(n;mathbbR)$ to $O(p,q)$?

I think this follows for groups $G$ abstractly isomorphic to $O(n)$ as this is the maximal compact subgroup and hence unique up to conjugacy. I'm looking for suggestions on how to go about proving (or looking for counterexamples) to the indefinite cases.

linear-algebra lie-groups

share|cite|improve this question

edited Mar 15 at 16:17

Steve

asked Mar 15 at 5:26

SteveSteve

61949

$endgroup$

share|cite|improve this question

edited Mar 15 at 16:17

Steve

asked Mar 15 at 5:26

SteveSteve

61949

share|cite|improve this question

edited Mar 15 at 16:17

Steve

asked Mar 15 at 5:26

SteveSteve

61949

asked Mar 15 at 5:26

SteveSteve

61949

asked Mar 15 at 5:26

SteveSteve

61949