What is the group of outer automorphisms of $SL_n(\mathbb{Z})$. I wanted to understand semidirect products of the form $SL_n(\mathbb{Z})\rtimes_\varphi \mathbb{Z}$ and its isomorphism type depends only on $[\varphi]\in Out(SL_n(\mathbb{Z})$. There is always the conjugate inverse, which is clearly not an inner automorphism, as it doesn't preserve the minimal polynomial of the matrix (at least for $n\ge 3$). Are there any outer other outer automorphisms ?
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Are all automorphisms What is the group of $SL_n(\mathbb{Z})$ inner outer automorphisms (for of $n\ge 3$)? SL_n(\mathbb{Z})$. I wanted to understand semidirect products of the form $SL_n(\mathbb{Z})\rtimes SL_n(\mathbb{Z})\rtimes_\varphi \mathbb{Z}$ and if there are no outer automophismsits isomorphism type depends only on $[\varphi]\in Out(SL_n(\mathbb{Z})$. There is always the conjugate inverse, then they would all be isomorphic to a direct productwhich is clearly not an inner automorphism, as it doesn't preserve the minimal polynomial of the matrix (at least for $n\ge 3$). Are there any outer outer automorphisms ? |
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Automorphisms of $SL_n(\mathbb{Z})$Are all automorphisms of $SL_n(\mathbb{Z})$ inner automorphisms (for $n\ge 3$)? I wanted to understand semidirect products of the form $SL_n(\mathbb{Z})\rtimes \mathbb{Z}$ and if there are no outer automophisms, then they would all be isomorphic to a direct product.
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