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For every finite group $G$ , does there exist $n \in \mathbb Z^+$ such that $G$ can be embedded in $SO_n(\mathbb R)$ ? Can every finite group be embedded in $SL_n(\mathbb R)$ for some $n$ ?

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Yes. $G$ acts on itself by left multiplication, which gives us an embedding as group of permutation matrices into $GL_{|G|}(\mathbb R)$. Clearly, permutation matrices are orthogonal and have determinant $\pm1$. By adding another dimension that "eats" the sign, we obtain an embedding $G\to GL_{|G|+1}(\mathbb R)$ where the image is in fact in $SL_{|G|+1}$ and $SO_{|G|+1}$.

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  • $\begingroup$ I thought of permutation matrices but I wasn't sure what to do about the sign though. Should it be obvious how to use such a "hungry" extra dimension? $\endgroup$
    – pjs36
    Aug 13 '15 at 16:34

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