References for (especially two-dimensional) general linear groups over *finite* fields

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Questions. What are good, citable, detailed sources on general linear groups over finite fields? Especially, $mathrmGL_2(mathbbF_p)$, and most especially, the following:



  • characterization of normalizers of tori in $mathrmGL_2(mathbbF_p)$, (I am sorry, but I can't get more specific about this rather special concern, for I want to play by the conventional rules of the reviewing process and not identify myself too much.)


  • detailed classification of when an element $AinmathrmGL_2(mathbbF_p)$ is diagonalizable. There seem to be detailed results on what field-extensions of $mathbbF_p$ the eigenvalues of diagonalizable $AinmathrmGL_2(mathbbF_p)$ can possibly lie in (this I infer from the assertions I have to check), but within the time I took during a reviewing job, I did not find a good reference. Algebraically closed fields dominate the literature, needless to say,


  • characterization of all elements of $mathrmPGL(2,mathbbF_p)$ which have order equal to $p$,


  • classification of the intersection of two distinct tori in $mathrmGL_2(mathbbF_p)$.


Remarks. Motivation for this reference requestion is that---in a mostly combinatorial article that I am tasked with reviewing---on one page, all of a sudden, I am confronted is a barrage of non-trivial (though no doubt known to experts) assertions about general linear groups over finite fields. (They are using these statements en route to another result.) It is strange, and somewhat reproachable, that the authors do not give a single reference for their statements, writing as if this was an article in a professional algebra journal.




reference-request finite-fields algebraic-groups linear-groups




share|cite|improve this question





















  • If you don't want to identify yourself, perhaps change your username ? (sorry I can't help you with the question though)
    – Max
    Jul 30 at 9:22










  • @Max: thanks for the suggestion; it's not that sensitive; my question is obfuscated enough; besides, there are circumstances which make this reviewing task a particularly relaxed matter. I'm not comfortable saying more. I am looking for a specialized reference on general linear, and projective linear, groups over finite fields. They are only a side show in a longer combinatorial paper though.
    – Peter Heinig
    Jul 30 at 10:54










  • Would you be interested in solutions to the issues you ask about, or only references?
    – KReiser
    Jul 30 at 20:57






  • 1




    Everything you want falls out of Jordan normal form, Cayley-Hamilton, and some very basic finite field theory. C-H gives that the eigenvalues (if they exist) are either both in $Bbb F_p$ or $Bbb F_p^2$, so the matrix is diagonalizable over $Bbb F_p^2$ iff it's diagonalizable. If not, it's similar to a Jordan block with diag entries in $Bbb F_p$. Nondiag'ble elts have order $p$, diag'ble over $Bbb F_p$ have order dividing $p-1$, and diag'ble elts over $Bbb F_p^2$ have order dividing $p^2-1$ but not $p-1$ because the Frobenius is the unique element of the Galois group...
    – KReiser
    Jul 31 at 18:08






  • 1




    ... and all of these matrix computations can be performed by hand without too much trouble. I think they're probably on MO or MSE somewhere, too. For the bits regarding tori, most introductory Lie theory books should have the classification of normalizers of (diagonal) tori in them, and those proofs should be able to be phrased without reliance on characteristic. I would also suggest that you ask your authors to include the citations for this in their paper, or failing that, include the results so that others may cite their paper.
    – KReiser
    Jul 31 at 18:13















up vote
1
down vote

favorite












Questions. What are good, citable, detailed sources on general linear groups over finite fields? Especially, $mathrmGL_2(mathbbF_p)$, and most especially, the following:



  • characterization of normalizers of tori in $mathrmGL_2(mathbbF_p)$, (I am sorry, but I can't get more specific about this rather special concern, for I want to play by the conventional rules of the reviewing process and not identify myself too much.)


  • detailed classification of when an element $AinmathrmGL_2(mathbbF_p)$ is diagonalizable. There seem to be detailed results on what field-extensions of $mathbbF_p$ the eigenvalues of diagonalizable $AinmathrmGL_2(mathbbF_p)$ can possibly lie in (this I infer from the assertions I have to check), but within the time I took during a reviewing job, I did not find a good reference. Algebraically closed fields dominate the literature, needless to say,


  • characterization of all elements of $mathrmPGL(2,mathbbF_p)$ which have order equal to $p$,


  • classification of the intersection of two distinct tori in $mathrmGL_2(mathbbF_p)$.


Remarks. Motivation for this reference requestion is that---in a mostly combinatorial article that I am tasked with reviewing---on one page, all of a sudden, I am confronted is a barrage of non-trivial (though no doubt known to experts) assertions about general linear groups over finite fields. (They are using these statements en route to another result.) It is strange, and somewhat reproachable, that the authors do not give a single reference for their statements, writing as if this was an article in a professional algebra journal.




reference-request finite-fields algebraic-groups linear-groups




share|cite|improve this question





















  • If you don't want to identify yourself, perhaps change your username ? (sorry I can't help you with the question though)
    – Max
    Jul 30 at 9:22










  • @Max: thanks for the suggestion; it's not that sensitive; my question is obfuscated enough; besides, there are circumstances which make this reviewing task a particularly relaxed matter. I'm not comfortable saying more. I am looking for a specialized reference on general linear, and projective linear, groups over finite fields. They are only a side show in a longer combinatorial paper though.
    – Peter Heinig
    Jul 30 at 10:54










  • Would you be interested in solutions to the issues you ask about, or only references?
    – KReiser
    Jul 30 at 20:57






  • 1




    Everything you want falls out of Jordan normal form, Cayley-Hamilton, and some very basic finite field theory. C-H gives that the eigenvalues (if they exist) are either both in $Bbb F_p$ or $Bbb F_p^2$, so the matrix is diagonalizable over $Bbb F_p^2$ iff it's diagonalizable. If not, it's similar to a Jordan block with diag entries in $Bbb F_p$. Nondiag'ble elts have order $p$, diag'ble over $Bbb F_p$ have order dividing $p-1$, and diag'ble elts over $Bbb F_p^2$ have order dividing $p^2-1$ but not $p-1$ because the Frobenius is the unique element of the Galois group...
    – KReiser
    Jul 31 at 18:08






  • 1




    ... and all of these matrix computations can be performed by hand without too much trouble. I think they're probably on MO or MSE somewhere, too. For the bits regarding tori, most introductory Lie theory books should have the classification of normalizers of (diagonal) tori in them, and those proofs should be able to be phrased without reliance on characteristic. I would also suggest that you ask your authors to include the citations for this in their paper, or failing that, include the results so that others may cite their paper.
    – KReiser
    Jul 31 at 18:13













up vote
1
down vote

favorite









up vote
1
down vote

favorite











Questions. What are good, citable, detailed sources on general linear groups over finite fields? Especially, $mathrmGL_2(mathbbF_p)$, and most especially, the following:



  • characterization of normalizers of tori in $mathrmGL_2(mathbbF_p)$, (I am sorry, but I can't get more specific about this rather special concern, for I want to play by the conventional rules of the reviewing process and not identify myself too much.)


  • detailed classification of when an element $AinmathrmGL_2(mathbbF_p)$ is diagonalizable. There seem to be detailed results on what field-extensions of $mathbbF_p$ the eigenvalues of diagonalizable $AinmathrmGL_2(mathbbF_p)$ can possibly lie in (this I infer from the assertions I have to check), but within the time I took during a reviewing job, I did not find a good reference. Algebraically closed fields dominate the literature, needless to say,


  • characterization of all elements of $mathrmPGL(2,mathbbF_p)$ which have order equal to $p$,


  • classification of the intersection of two distinct tori in $mathrmGL_2(mathbbF_p)$.


Remarks. Motivation for this reference requestion is that---in a mostly combinatorial article that I am tasked with reviewing---on one page, all of a sudden, I am confronted is a barrage of non-trivial (though no doubt known to experts) assertions about general linear groups over finite fields. (They are using these statements en route to another result.) It is strange, and somewhat reproachable, that the authors do not give a single reference for their statements, writing as if this was an article in a professional algebra journal.




reference-request finite-fields algebraic-groups linear-groups




share|cite|improve this question













Questions. What are good, citable, detailed sources on general linear groups over finite fields? Especially, $mathrmGL_2(mathbbF_p)$, and most especially, the following:



  • characterization of normalizers of tori in $mathrmGL_2(mathbbF_p)$, (I am sorry, but I can't get more specific about this rather special concern, for I want to play by the conventional rules of the reviewing process and not identify myself too much.)


  • detailed classification of when an element $AinmathrmGL_2(mathbbF_p)$ is diagonalizable. There seem to be detailed results on what field-extensions of $mathbbF_p$ the eigenvalues of diagonalizable $AinmathrmGL_2(mathbbF_p)$ can possibly lie in (this I infer from the assertions I have to check), but within the time I took during a reviewing job, I did not find a good reference. Algebraically closed fields dominate the literature, needless to say,


  • characterization of all elements of $mathrmPGL(2,mathbbF_p)$ which have order equal to $p$,


  • classification of the intersection of two distinct tori in $mathrmGL_2(mathbbF_p)$.


Remarks. Motivation for this reference requestion is that---in a mostly combinatorial article that I am tasked with reviewing---on one page, all of a sudden, I am confronted is a barrage of non-trivial (though no doubt known to experts) assertions about general linear groups over finite fields. (They are using these statements en route to another result.) It is strange, and somewhat reproachable, that the authors do not give a single reference for their statements, writing as if this was an article in a professional algebra journal.





reference-request finite-fields algebraic-groups linear-groups





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 1 at 9:08
























asked Jul 30 at 9:17









Peter Heinig

867316




867316











  • If you don't want to identify yourself, perhaps change your username ? (sorry I can't help you with the question though)
    – Max
    Jul 30 at 9:22










  • @Max: thanks for the suggestion; it's not that sensitive; my question is obfuscated enough; besides, there are circumstances which make this reviewing task a particularly relaxed matter. I'm not comfortable saying more. I am looking for a specialized reference on general linear, and projective linear, groups over finite fields. They are only a side show in a longer combinatorial paper though.
    – Peter Heinig
    Jul 30 at 10:54










  • Would you be interested in solutions to the issues you ask about, or only references?
    – KReiser
    Jul 30 at 20:57






  • 1




    Everything you want falls out of Jordan normal form, Cayley-Hamilton, and some very basic finite field theory. C-H gives that the eigenvalues (if they exist) are either both in $Bbb F_p$ or $Bbb F_p^2$, so the matrix is diagonalizable over $Bbb F_p^2$ iff it's diagonalizable. If not, it's similar to a Jordan block with diag entries in $Bbb F_p$. Nondiag'ble elts have order $p$, diag'ble over $Bbb F_p$ have order dividing $p-1$, and diag'ble elts over $Bbb F_p^2$ have order dividing $p^2-1$ but not $p-1$ because the Frobenius is the unique element of the Galois group...
    – KReiser
    Jul 31 at 18:08






  • 1




    ... and all of these matrix computations can be performed by hand without too much trouble. I think they're probably on MO or MSE somewhere, too. For the bits regarding tori, most introductory Lie theory books should have the classification of normalizers of (diagonal) tori in them, and those proofs should be able to be phrased without reliance on characteristic. I would also suggest that you ask your authors to include the citations for this in their paper, or failing that, include the results so that others may cite their paper.
    – KReiser
    Jul 31 at 18:13

















  • If you don't want to identify yourself, perhaps change your username ? (sorry I can't help you with the question though)
    – Max
    Jul 30 at 9:22










  • @Max: thanks for the suggestion; it's not that sensitive; my question is obfuscated enough; besides, there are circumstances which make this reviewing task a particularly relaxed matter. I'm not comfortable saying more. I am looking for a specialized reference on general linear, and projective linear, groups over finite fields. They are only a side show in a longer combinatorial paper though.
    – Peter Heinig
    Jul 30 at 10:54










  • Would you be interested in solutions to the issues you ask about, or only references?
    – KReiser
    Jul 30 at 20:57






  • 1




    Everything you want falls out of Jordan normal form, Cayley-Hamilton, and some very basic finite field theory. C-H gives that the eigenvalues (if they exist) are either both in $Bbb F_p$ or $Bbb F_p^2$, so the matrix is diagonalizable over $Bbb F_p^2$ iff it's diagonalizable. If not, it's similar to a Jordan block with diag entries in $Bbb F_p$. Nondiag'ble elts have order $p$, diag'ble over $Bbb F_p$ have order dividing $p-1$, and diag'ble elts over $Bbb F_p^2$ have order dividing $p^2-1$ but not $p-1$ because the Frobenius is the unique element of the Galois group...
    – KReiser
    Jul 31 at 18:08






  • 1




    ... and all of these matrix computations can be performed by hand without too much trouble. I think they're probably on MO or MSE somewhere, too. For the bits regarding tori, most introductory Lie theory books should have the classification of normalizers of (diagonal) tori in them, and those proofs should be able to be phrased without reliance on characteristic. I would also suggest that you ask your authors to include the citations for this in their paper, or failing that, include the results so that others may cite their paper.
    – KReiser
    Jul 31 at 18:13
















If you don't want to identify yourself, perhaps change your username ? (sorry I can't help you with the question though)
– Max
Jul 30 at 9:22




If you don't want to identify yourself, perhaps change your username ? (sorry I can't help you with the question though)
– Max
Jul 30 at 9:22












@Max: thanks for the suggestion; it's not that sensitive; my question is obfuscated enough; besides, there are circumstances which make this reviewing task a particularly relaxed matter. I'm not comfortable saying more. I am looking for a specialized reference on general linear, and projective linear, groups over finite fields. They are only a side show in a longer combinatorial paper though.
– Peter Heinig
Jul 30 at 10:54




@Max: thanks for the suggestion; it's not that sensitive; my question is obfuscated enough; besides, there are circumstances which make this reviewing task a particularly relaxed matter. I'm not comfortable saying more. I am looking for a specialized reference on general linear, and projective linear, groups over finite fields. They are only a side show in a longer combinatorial paper though.
– Peter Heinig
Jul 30 at 10:54












Would you be interested in solutions to the issues you ask about, or only references?
– KReiser
Jul 30 at 20:57




Would you be interested in solutions to the issues you ask about, or only references?
– KReiser
Jul 30 at 20:57




1




1




Everything you want falls out of Jordan normal form, Cayley-Hamilton, and some very basic finite field theory. C-H gives that the eigenvalues (if they exist) are either both in $Bbb F_p$ or $Bbb F_p^2$, so the matrix is diagonalizable over $Bbb F_p^2$ iff it's diagonalizable. If not, it's similar to a Jordan block with diag entries in $Bbb F_p$. Nondiag'ble elts have order $p$, diag'ble over $Bbb F_p$ have order dividing $p-1$, and diag'ble elts over $Bbb F_p^2$ have order dividing $p^2-1$ but not $p-1$ because the Frobenius is the unique element of the Galois group...
– KReiser
Jul 31 at 18:08




Everything you want falls out of Jordan normal form, Cayley-Hamilton, and some very basic finite field theory. C-H gives that the eigenvalues (if they exist) are either both in $Bbb F_p$ or $Bbb F_p^2$, so the matrix is diagonalizable over $Bbb F_p^2$ iff it's diagonalizable. If not, it's similar to a Jordan block with diag entries in $Bbb F_p$. Nondiag'ble elts have order $p$, diag'ble over $Bbb F_p$ have order dividing $p-1$, and diag'ble elts over $Bbb F_p^2$ have order dividing $p^2-1$ but not $p-1$ because the Frobenius is the unique element of the Galois group...
– KReiser
Jul 31 at 18:08




1




1




... and all of these matrix computations can be performed by hand without too much trouble. I think they're probably on MO or MSE somewhere, too. For the bits regarding tori, most introductory Lie theory books should have the classification of normalizers of (diagonal) tori in them, and those proofs should be able to be phrased without reliance on characteristic. I would also suggest that you ask your authors to include the citations for this in their paper, or failing that, include the results so that others may cite their paper.
– KReiser
Jul 31 at 18:13





... and all of these matrix computations can be performed by hand without too much trouble. I think they're probably on MO or MSE somewhere, too. For the bits regarding tori, most introductory Lie theory books should have the classification of normalizers of (diagonal) tori in them, and those proofs should be able to be phrased without reliance on characteristic. I would also suggest that you ask your authors to include the citations for this in their paper, or failing that, include the results so that others may cite their paper.
– KReiser
Jul 31 at 18:13
















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