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T.Ch.
T.Ch.
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$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Assume that we have two subgroups $G_1,G_2$ of $\Sp(4,\mathbb{Z})$ that are conjugate in $\GL(4,\mathbb{Q})$ $\big($or in $\SL(4,\mathbb{Z})$ if that helps$\big)$.

Does it followsfollow that their indices are equal: $[\Sp(4,\mathbb{Z}):G_1]=[\Sp(4,\mathbb{Z}):G_2]$? If not, is it at least true that if one index is finite, then so is the other?

Thanks!

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Assume that we have two subgroups $G_1,G_2$ of $\Sp(4,\mathbb{Z})$ that are conjugate in $\GL(4,\mathbb{Q})$ $\big($or in $\SL(4,\mathbb{Z})$ if that helps$\big)$.

Does it follows that their indices are equal: $[\Sp(4,\mathbb{Z}):G_1]=[\Sp(4,\mathbb{Z}):G_2]$? If not, is it at least true that if one index is finite, then so is the other?

Thanks!

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Assume that we have two subgroups $G_1,G_2$ of $\Sp(4,\mathbb{Z})$ that are conjugate in $\GL(4,\mathbb{Q})$ $\big($or in $\SL(4,\mathbb{Z})$ if that helps$\big)$.

Does it follow that their indices are equal: $[\Sp(4,\mathbb{Z}):G_1]=[\Sp(4,\mathbb{Z}):G_2]$? If not, is it at least true that if one index is finite, then so is the other?

Thanks!

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YCor
YCor
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Index of subgroups of $Sp$\mathrm{Sp}(4,\mathbb{Z})$ conjugate in $GL$\mathrm{GL}(4,\mathbb{Q})$

Assume$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Assume that we have two subgroups $G_1,G_2$ of $Sp(4,\mathbb{Z})$$\Sp(4,\mathbb{Z})$ that are conjugate in $GL(4,\mathbb{Q})$$\GL(4,\mathbb{Q})$ $\big($or in $SL(4,\mathbb{Z})$$\SL(4,\mathbb{Z})$ if that helps$\big)$.

Does it follows that their indices are equal: $[Sp(4,\mathbb{Z}):G_1]=[Sp(4,\mathbb{Z}):G_2]$$[\Sp(4,\mathbb{Z}):G_1]=[\Sp(4,\mathbb{Z}):G_2]$? If not, is it at least true that if one index is finite, then so is the other?

Thanks!

Index of subgroups of $Sp(4,\mathbb{Z})$ conjugate in $GL(4,\mathbb{Q})$

Assume that we have two subgroups $G_1,G_2$ of $Sp(4,\mathbb{Z})$ that are conjugate in $GL(4,\mathbb{Q})$ $\big($or in $SL(4,\mathbb{Z})$ if that helps$\big)$.

Does it follows that their indices are equal: $[Sp(4,\mathbb{Z}):G_1]=[Sp(4,\mathbb{Z}):G_2]$? If not, is it at least true that if one index is finite, then so is the other?

Thanks!

Index of subgroups of $\mathrm{Sp}(4,\mathbb{Z})$ conjugate in $\mathrm{GL}(4,\mathbb{Q})$

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Assume that we have two subgroups $G_1,G_2$ of $\Sp(4,\mathbb{Z})$ that are conjugate in $\GL(4,\mathbb{Q})$ $\big($or in $\SL(4,\mathbb{Z})$ if that helps$\big)$.

Does it follows that their indices are equal: $[\Sp(4,\mathbb{Z}):G_1]=[\Sp(4,\mathbb{Z}):G_2]$? If not, is it at least true that if one index is finite, then so is the other?

Thanks!

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T.Ch.
T.Ch.
  • 141
  • 4

Index of subgroups of $Sp(4,\mathbb{Z})$ conjugate in $GL(4,\mathbb{Q})$

Assume that we have two subgroups $G_1,G_2$ of $Sp(4,\mathbb{Z})$ that are conjugate in $GL(4,\mathbb{Q})$ $\big($or in $SL(4,\mathbb{Z})$ if that helps$\big)$.

Does it follows that their indices are equal: $[Sp(4,\mathbb{Z}):G_1]=[Sp(4,\mathbb{Z}):G_2]$? If not, is it at least true that if one index is finite, then so is the other?

Thanks!