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JGis
JGis
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I need a reference for the proof that the complex orthogonal group $SO_n($$SO_{2n+1}($ℂ$) = \{A\in GL_n($$) = \{A\in SL_{2n+1}($ℂ$): A^TA = Id\}$ is simple in a group theoretical sense (if it is true). How about the simplicity of $SO_n(K)$$SO_{2n+1}(K)$ in general (i.e. $K$ an arbitrary infinite field)? It there any criterion? It seems that if $K$ is "big", then $SO_n(K)$$SO_{2n+1}(K)$ is not simple, e.g. if $K$ has a valuation, then (I think) one can find somehow a proper normal subgroup.

I need a reference for the proof that the complex orthogonal group $SO_n($ℂ$) = \{A\in GL_n($ℂ$): A^TA = Id\}$ is simple in a group theoretical sense (if it is true). How about the simplicity of $SO_n(K)$ in general (i.e. $K$ an arbitrary infinite field)? It there any criterion? It seems that if $K$ is "big", then $SO_n(K)$ is not simple, e.g. if $K$ has a valuation, then (I think) one can find somehow a proper normal subgroup.

I need a reference for the proof that the complex orthogonal group $SO_{2n+1}($ℂ$) = \{A\in SL_{2n+1}($ℂ$): A^TA = Id\}$ is simple in a group theoretical sense (if it is true). How about the simplicity of $SO_{2n+1}(K)$ in general (i.e. $K$ an arbitrary infinite field)? It there any criterion? It seems that if $K$ is "big", then $SO_{2n+1}(K)$ is not simple, e.g. if $K$ has a valuation, then (I think) one can find somehow a proper normal subgroup.

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JGis
JGis
  • 331
  • 1
  • 9

Simplicity of (complex) orthogonal groups

I need a reference for the proof that the complex orthogonal group $SO_n($ℂ$) = \{A\in GL_n($ℂ$): A^TA = Id\}$ is simple in a group theoretical sense (if it is true). How about the simplicity of $SO_n(K)$ in general (i.e. $K$ an arbitrary infinite field)? It there any criterion? It seems that if $K$ is "big", then $SO_n(K)$ is not simple, e.g. if $K$ has a valuation, then (I think) one can find somehow a proper normal subgroup.