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David A. Craven
David A. Craven
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One can consult the tables of Bray-Holt-Roney Dougal to work out the subgroups of $\mathrm{PSU}_m$.

  1. For $m=5$, we have copies of $PSL_2(11)$ and $PSU_4(4)$.
  2. For $m=6$ we have $PSL_2(11)$, $PSL_3(4)$ and $PSU_4(9)$.
  3. For $m=7$ we have $PSU_3(9)$.
  4. For $m=8$ we have $PSL_3(4)$.
  5. For $m=9$ we have $PSL_2(19)$.
  6. For $m=10$ we have $PSL_2(19)$ and $PSL_3(4)$.
  7. For $m=11$ we have $PSL_2(23)$ and $PSU_5(4)$.
  8. For $m=12$ we have $PSL_2(23)$ and $PSL_2(9)$.

So far, so good. Bray-Holt-Roney Dougal stops at dimension 12. For dimensions 13 to 15 one consults tables in Anna Schoeder'sSchroeder's PhD thesis (St Andrews, 2015) and there are examples in those dimensions. For dimensions 16516 and 17 one consults the thesis of Daniel Rogers (Warwick, 2017). For 16 there is an example, but there are no interesting subgroups (i.e., finite subgroups not contained in a closed positive-dimensional subgroup) of $\mathrm{PSU}_{17}$ at all, never mind linear or unitary ones.

So it seems $m=17$ is the first case where things go wrong.

One can consult the tables of Bray-Holt-Roney Dougal to work out the subgroups of $\mathrm{PSU}_m$.

  1. For $m=5$, we have copies of $PSL_2(11)$ and $PSU_4(4)$.
  2. For $m=6$ we have $PSL_2(11)$, $PSL_3(4)$ and $PSU_4(9)$.
  3. For $m=7$ we have $PSU_3(9)$.
  4. For $m=8$ we have $PSL_3(4)$.
  5. For $m=9$ we have $PSL_2(19)$.
  6. For $m=10$ we have $PSL_2(19)$ and $PSL_3(4)$.
  7. For $m=11$ we have $PSL_2(23)$ and $PSU_5(4)$.
  8. For $m=12$ we have $PSL_2(23)$ and $PSL_2(9)$.

So far, so good. Bray-Holt-Roney Dougal stops at dimension 12. For dimensions 13 to 15 one consults tables in Anna Schoeder's PhD thesis (St Andrews, 2015) and there are examples in those dimensions. For dimensions 165 and 17 one consults the thesis of Daniel Rogers (Warwick, 2017). For 16 there is an example, but there are no interesting subgroups (i.e., finite subgroups not contained in a closed positive-dimensional subgroup) of $\mathrm{PSU}_{17}$ at all, never mind linear or unitary ones.

So it seems $m=17$ is the first case where things go wrong.

One can consult the tables of Bray-Holt-Roney Dougal to work out the subgroups of $\mathrm{PSU}_m$.

  1. For $m=5$, we have copies of $PSL_2(11)$ and $PSU_4(4)$.
  2. For $m=6$ we have $PSL_2(11)$, $PSL_3(4)$ and $PSU_4(9)$.
  3. For $m=7$ we have $PSU_3(9)$.
  4. For $m=8$ we have $PSL_3(4)$.
  5. For $m=9$ we have $PSL_2(19)$.
  6. For $m=10$ we have $PSL_2(19)$ and $PSL_3(4)$.
  7. For $m=11$ we have $PSL_2(23)$ and $PSU_5(4)$.
  8. For $m=12$ we have $PSL_2(23)$ and $PSL_2(9)$.

So far, so good. Bray-Holt-Roney Dougal stops at dimension 12. For dimensions 13 to 15 one consults tables in Anna Schroeder's PhD thesis (St Andrews, 2015) and there are examples in those dimensions. For dimensions 16 and 17 one consults the thesis of Daniel Rogers (Warwick, 2017). For 16 there is an example, but there are no interesting subgroups (i.e., finite subgroups not contained in a closed positive-dimensional subgroup) of $\mathrm{PSU}_{17}$ at all, never mind linear or unitary ones.

So it seems $m=17$ is the first case where things go wrong.

Source Link
David A. Craven
David A. Craven
  • 1.1k
  • 7
  • 13

One can consult the tables of Bray-Holt-Roney Dougal to work out the subgroups of $\mathrm{PSU}_m$.

  1. For $m=5$, we have copies of $PSL_2(11)$ and $PSU_4(4)$.
  2. For $m=6$ we have $PSL_2(11)$, $PSL_3(4)$ and $PSU_4(9)$.
  3. For $m=7$ we have $PSU_3(9)$.
  4. For $m=8$ we have $PSL_3(4)$.
  5. For $m=9$ we have $PSL_2(19)$.
  6. For $m=10$ we have $PSL_2(19)$ and $PSL_3(4)$.
  7. For $m=11$ we have $PSL_2(23)$ and $PSU_5(4)$.
  8. For $m=12$ we have $PSL_2(23)$ and $PSL_2(9)$.

So far, so good. Bray-Holt-Roney Dougal stops at dimension 12. For dimensions 13 to 15 one consults tables in Anna Schoeder's PhD thesis (St Andrews, 2015) and there are examples in those dimensions. For dimensions 165 and 17 one consults the thesis of Daniel Rogers (Warwick, 2017). For 16 there is an example, but there are no interesting subgroups (i.e., finite subgroups not contained in a closed positive-dimensional subgroup) of $\mathrm{PSU}_{17}$ at all, never mind linear or unitary ones.

So it seems $m=17$ is the first case where things go wrong.