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YCor
YCor
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Finite order elements of $GL_d$\mathrm{GL}_d(\mathbb{Z})$ that are conjugate to powers of themselves

Let$\DeclareMathOperator\GL{GL}$Let $A\in GL_d(\mathbb{Z})$$A\in \GL_d(\mathbb{Z})$ have finite order $n.$ Suppose that $k\in \mathbb{Z}$ is relatively prime to $n.$ Is $A^k$ conjugate to $A$ in $GL_d(\mathbb{Z})$$\GL_d(\mathbb{Z})$?

For $d\leq 4$ the answer is yes. Indeed the papers "On the finite subgroups of $GL(3,\mathbb{Z})$""On the finite subgroups of $\GL(3,\mathbb{Z})$" by K. Tahara, 1971 and "Conjugacy Classes of Torsion in $GL_n(\mathbb{Z})$" "Conjugacy Classes of Torsion in $\GL_n(\mathbb{Z})$", by Q. Yang, 2015 list all torsion elements up to conjugacy for the cases $d=2,3$ and $d=4$ respectively. There are not many cases that need to be checked, so I checked each case and the answer turns out to be "yes" in all of these cases. But I have no general argument for why things work out when $d\leq 4$, only computations.

Finite order elements of $GL_d(\mathbb{Z})$ that are conjugate to powers of themselves

Let $A\in GL_d(\mathbb{Z})$ have finite order $n.$ Suppose that $k\in \mathbb{Z}$ is relatively prime to $n.$ Is $A^k$ conjugate to $A$ in $GL_d(\mathbb{Z})$?

For $d\leq 4$ the answer is yes. Indeed the papers "On the finite subgroups of $GL(3,\mathbb{Z})$" by K. Tahara, 1971 and "Conjugacy Classes of Torsion in $GL_n(\mathbb{Z})$", by Q. Yang, 2015 list all torsion elements up to conjugacy for the cases $d=2,3$ and $d=4$ respectively. There are not many cases that need to be checked, so I checked each case and the answer turns out to be "yes" in all of these cases. But I have no general argument for why things work out when $d\leq 4$, only computations.

Finite order elements of $\mathrm{GL}_d(\mathbb{Z})$ that are conjugate to powers of themselves

$\DeclareMathOperator\GL{GL}$Let $A\in \GL_d(\mathbb{Z})$ have finite order $n.$ Suppose that $k\in \mathbb{Z}$ is relatively prime to $n.$ Is $A^k$ conjugate to $A$ in $\GL_d(\mathbb{Z})$?

For $d\leq 4$ the answer is yes. Indeed the papers "On the finite subgroups of $\GL(3,\mathbb{Z})$" by K. Tahara, 1971 and "Conjugacy Classes of Torsion in $\GL_n(\mathbb{Z})$", by Q. Yang, 2015 list all torsion elements up to conjugacy for the cases $d=2,3$ and $d=4$ respectively. There are not many cases that need to be checked, so I checked each case and the answer turns out to be "yes" in all of these cases. But I have no general argument for why things work out when $d\leq 4$, only computations.

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YCor
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Caleb Eckhardt
Caleb Eckhardt
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Finite order elements of $GL_d(\mathbb{Z})$ that are conjugate to powers of themselves

Let $A\in GL_d(\mathbb{Z})$ have finite order $n.$ Suppose that $k\in \mathbb{Z}$ is relatively prime to $n.$ Is $A^k$ conjugate to $A$ in $GL_d(\mathbb{Z})$?

For $d\leq 4$ the answer is yes. Indeed the papers "On the finite subgroups of $GL(3,\mathbb{Z})$" by K. Tahara, 1971 and "Conjugacy Classes of Torsion in $GL_n(\mathbb{Z})$", by Q. Yang, 2015 list all torsion elements up to conjugacy for the cases $d=2,3$ and $d=4$ respectively. There are not many cases that need to be checked, so I checked each case and the answer turns out to be "yes" in all of these cases. But I have no general argument for why things work out when $d\leq 4$, only computations.