Is SO(n) a maximal subgroup of SO(n+1)?
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It is properly contained in $S(O(n)\times O(1))$.
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$\begingroup$ A check by hand on the Lie algebras shows that it is not contained in a proper subgroup of bigger dimension. $\endgroup$ Commented Jul 15, 2015 at 11:24
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$\begingroup$ Can you tell me what you mean by S(O(n)×O(1))? $\endgroup$– hosainCommented Jul 15, 2015 at 14:14
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$\begingroup$ This means block matrices $\begin{pmatrix}A&0\\0&B\end{pmatrix}$ where $A\in O(n),B\in O(1)$ and the whole matrix has determinant 1. $\endgroup$ Commented Jul 15, 2015 at 14:17
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$\begingroup$ @ user75905 How can you prove it? $\endgroup$– hosainCommented Jul 26, 2015 at 18:36