Timeline for Simultaneous similarity classes of pairs in $\mathrm{GL}_{n}(\Bbb Z / p\Bbb Z)$?
Current License: CC BY-SA 4.0
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| Nov 13, 2020 at 18:01 | history | edited | Nourr Mga |
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| Nov 11, 2020 at 20:31 | history | edited | Nourr Mga | CC BY-SA 4.0 |
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| Nov 10, 2020 at 12:37 | comment | added | Nourr Mga | Ok, thank you. Maybe I didn't understand it very well, or I missed some think. what do you think about the second question?. | |
| Nov 9, 2020 at 22:58 | comment | added | YCor | No, it doesn't, you should read Derek's comment more carefully. | |
| Nov 9, 2020 at 22:26 | comment | added | Nourr Mga | @Derek Holt, if $\alpha$ and $\beta$ are similar, it follows that $\alpha(G)$ and $\beta(G)$ are conjugate subgroup in $ GL_{n}(\Bbb Z / p\Bbb Z)$ which led to the simultaneous similarity of their generators. | |
| Nov 9, 2020 at 22:07 | history | edited | Nourr Mga | CC BY-SA 4.0 |
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| Nov 9, 2020 at 22:01 | history | edited | Nourr Mga | CC BY-SA 4.0 |
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| Nov 9, 2020 at 21:59 | comment | added | YCor | So you're using $A$ for two different things; it seems that $A$ denotes $\mathrm{GL}_2(\mathbf{Z}/p\mathbf{Z})$ at the beginning, and the image of the first generator (hence a given element of $\mathrm{GL}_2(\mathbf{Z}/p\mathbf{Z})$) later. Why don't you fix it? | |
| Nov 9, 2020 at 21:58 | comment | added | Nourr Mga | I added some clarifications to the question. @YCor of course I m not loking to A as image of the first generator of G. In fact, if $\alpha$ and $\beta$ are similar it follows that $\alpha(G)$ and $\beta(G)$ are conjugate subgroup which led to the simultaneous similarity of their generators. | |
| Nov 9, 2020 at 21:41 | history | edited | Nourr Mga | CC BY-SA 4.0 |
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| Nov 9, 2020 at 13:36 | comment | added | Derek Holt | I am not sure whether I am understanding the definitions properly, and you need to sort out your notation, but it looks as thought the pairs $(A,B)$ and $(B,A)$ could fail to be simultaneously similar, but the corresponding homomorphisms $\alpha, \beta$ would be similar, because there is an automorphism of $\langle A,B \rangle$ that swaps $A$ and $B$. | |
| Nov 9, 2020 at 12:08 | comment | added | YCor | But what is $A$ in $\mathrm{Aut}(A)$. Probably not the image of the first generator, also denoted $A$. | |
| Nov 9, 2020 at 11:35 | comment | added | Nourddine Snanou | @YCor. You're right. A, B are the images of two given generators of G. | |
| Nov 9, 2020 at 4:12 | comment | added | YCor | It seems implicit that $A,B$ are the images of two given generators of $G$ (rather than an arbitrary generating pair of $\alpha(G)$, as the current notation seems to indicate)? Also $A$ is used to denote a group, and then to denote a matrix. | |
| Nov 9, 2020 at 4:10 | history | edited | YCor | CC BY-SA 4.0 |
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| Nov 9, 2020 at 1:17 | history | edited | Nourr Mga | CC BY-SA 4.0 |
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| Nov 9, 2020 at 1:16 | history | undeleted | Nourr Mga | ||
| Nov 9, 2020 at 0:52 | history | deleted | Nourr Mga | via Vote | |
| Nov 9, 2020 at 0:51 | history | asked | Nourr Mga | CC BY-SA 4.0 |