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Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$. We also assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\langle G,g\rangle}=\mathrm{SO}_n(\mathbb R)$. Let $H:=\{h\in\mathrm{O}_n(\mathbb R)\setminus\mathrm{SO}_n(\mathbb R): |\langle G,h\rangle|<\infty\}$. Is it true that for any $h,h'\in H$, if $h\overset{\mathrm{O}_n(\mathbb R)}\sim h'$, then $\langle G,h\rangle\overset{\mathrm{O}_n(\mathbb R)}\sim\langle G,h'\rangle$?

Where $\overset{\mathrm{O}_n(\mathbb R)}\sim$ is the conjugate relation in $\mathrm{O}_n(\mathbb R)$. Notice that for any $h\in H$, $|\langle G,h\rangle|=2|G|$, because $G=\ker\det$ is a normal subgroup of $\langle G,h\rangle$ of index 2.

Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$. We also assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\langle G,g\rangle}=\mathrm{SO}_n(\mathbb R)$. Let $H:=\{h\in\mathrm{O}_n(\mathbb R)\setminus\mathrm{SO}_n(\mathbb R): |\langle G,h\rangle|<\infty\}$. Is it true that for any $h,h'\in H$, if $h\overset{\mathrm{O}_n(\mathbb R)}\sim h'$, then $\langle G,h\rangle\overset{\mathrm{O}_n(\mathbb R)}\sim\langle G,h'\rangle,$ where $\overset{\mathrm{O}_n(\mathbb R)}\sim$ is the conjugate relation in $\mathrm{O}_n(\mathbb R)$? Notice that for any $h\in H$, $|\langle G,h\rangle|=2|G|$, because $G=\ker\det$ is a normal subgroup of $\langle G,h\rangle$ of index $2.$

Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$ and let $H:=\{h\in\mathrm{O}_n(\mathbb R)\setminus\mathrm{SO}_n(\mathbb R): |\langle G,h\rangle|<\infty\}$. Is it true that for any $h,h'\in H$, if $h\overset{\mathrm{O}_n(\mathbb R)}\sim h'$, then $\langle G,h\rangle\overset{\mathrm{O}_n(\mathbb R)}\sim\langle G,h'\rangle$?

Where $\overset{\mathrm{O}_n(\mathbb R)}\sim$ is the conjugate relation in $\mathrm{O}_n(\mathbb R)$. Notice that for any $h\in H$, $|\langle G,h\rangle|=2|G|$, because $G=\ker\det$ is a normal subgroup of $\langle G,h\rangle$ of index 2. If this makes any difference, we can assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\langle G,g\rangle}=\mathrm{SO}_n(\mathbb R)$.

Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$. We also assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\langle G,g\rangle}=\mathrm{SO}_n(\mathbb R)$. Let $H:=\{h\in\mathrm{O}_n(\mathbb R)\setminus\mathrm{SO}_n(\mathbb R): |\langle G,h\rangle|<\infty\}$. Is it true that for any $h,h'\in H$, if $h\overset{\mathrm{O}_n(\mathbb R)}\sim h'$, then $\langle G,h\rangle\overset{\mathrm{O}_n(\mathbb R)}\sim\langle G,h'\rangle$?

Where $\overset{\mathrm{O}_n(\mathbb R)}\sim$ is the conjugate relation in $\mathrm{O}_n(\mathbb R)$. Notice that for any $h\in H$, $|\langle G,h\rangle|=2|G|$, because $G=\ker\det$ is a normal subgroup of $\langle G,h\rangle$ of index 2.

Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$ and let $H:=\{h\in\mathrm{O}_n(\mathbb R)\setminus\mathrm{SO}_n(\mathbb R): |\langle G,h\rangle|<\infty\}$. Is true that for any $h,h'\in H$, if $h\overset{\mathrm{O}_n(\mathbb R)}\sim h'$, then $\langle G,h\rangle\overset{\mathrm{O}_n(\mathbb R)}\sim\langle G,h'\rangle$?

Where $\overset{\mathrm{O}_n(\mathbb R)}\sim$ is the conjugate relation in $\mathrm{O}_n(\mathbb R)$. Notice that for any $h\in H$, $|\langle G,h\rangle|=2|G|$, because $G=\ker\det$ is a normal subgroup of $\langle G,h\rangle$ of index 2. If this makes any difference, we can assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\langle G,g\rangle}=\mathrm{SO}_n(\mathbb R)$.

Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$ and let $H:=\{h\in\mathrm{O}_n(\mathbb R)\setminus\mathrm{SO}_n(\mathbb R): |\langle G,h\rangle|<\infty\}$. Is it true that for any $h,h'\in H$, if $h\overset{\mathrm{O}_n(\mathbb R)}\sim h'$, then $\langle G,h\rangle\overset{\mathrm{O}_n(\mathbb R)}\sim\langle G,h'\rangle$?

Where $\overset{\mathrm{O}_n(\mathbb R)}\sim$ is the conjugate relation in $\mathrm{O}_n(\mathbb R)$. Notice that for any $h\in H$, $|\langle G,h\rangle|=2|G|$, because $G=\ker\det$ is a normal subgroup of $\langle G,h\rangle$ of index 2. If this makes any difference, we can assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\langle G,g\rangle}=\mathrm{SO}_n(\mathbb R)$.