- Give an interesting example of a non-normal subgroup in a familiar group.
- Is the rotation subgroup \(\operatorname{SO}(2)\) normal inside the special linear group \(\operatorname{SL}_2(\mathbb{R})\)? Prove your assertion with an explicit calculation.
Solution
A familiar example is \[ \langle(12)\rangle\le S_3. \] It is not normal because \[ (123)(12)(123)^{-1}=(23)\notin\langle(12)\rangle. \]
Likewise, \(\operatorname{SO}(2)\) is not normal in \(\operatorname{SL}_2(\mathbb R)\). Let \[ J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\in\operatorname{SO}(2), \qquad g=\begin{pmatrix}2&0\\0&1/2\end{pmatrix}\in\operatorname{SL}_2(\mathbb R). \] Then \[ gJg^{-1} =\begin{pmatrix}0&-4\\1/4&0\end{pmatrix}. \] Its first column has Euclidean norm \(1/4\), so it is not orthogonal. Hence \[ gJg^{-1}\notin\operatorname{SO}(2), \] which proves \[ \operatorname{SO}(2)\not\trianglelefteq\operatorname{SL}_2(\mathbb R). \]