Timeline for normalizer quotient is $\operatorname{GL}_2(p)$
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 23, 2023 at 1:03 | comment | added | user488802 | Sorry to trouble with a related question. Could I also conclude that the conjugacy class in $G$ to which $E$ belongs is unique because it is from the regular representation? | |
| Jun 22, 2023 at 8:52 | comment | added | user488802 | I think I've got it. Thank you for your patience! | |
| Jun 22, 2023 at 7:41 | comment | added | Dave Benson | Okay, let me express this in terms of character theory. Your condition implies that the trace of all non-identity elements of $E$ is zero. So the inner products of your character with all one dimrensional irreducibles are equal. So the number of copies of each is the same. It follows that all automorphisms of $E$ take this to an isomorphic representation. The isomorphism then gives the required element of the normaliser to effect this automorphism of $E$. I hope this helps. | |
| Jun 22, 2023 at 4:23 | comment | added | user488802 | Because there are other reps (where not every nontrivial elt is conjugate to $e$) which don't give this $N/C$ structure. | |
| Jun 22, 2023 at 4:18 | comment | added | user488802 | I suppose my question is: for the 2nd possibility, why this particular representation where all the nontrivial elts are conjugate to $e$ gives the $N/C$ structure? | |
| Jun 22, 2023 at 0:01 | comment | added | user488802 | I seem to have grasped your answer, thank you! Just one question on the last sentence. I'm being stupid here. Why in the regular representation case, $N/C$ is the full automorphism group? And you mean $k/p$ copies of the regular reps, right? Thank you! | |
| Jun 21, 2023 at 22:58 | vote | accept | user488802 | ||
| Jun 21, 2023 at 22:34 | comment | added | user488802 | Sorry about that... | |
| Jun 21, 2023 at 22:33 | comment | added | Dave Benson | You also keep accepting and unaccepting my answer, which is a little annoying. | |
| Jun 21, 2023 at 22:32 | comment | added | user488802 | Sorry I forgot to write "thank you for all your input". | |
| Jun 21, 2023 at 22:09 | vote | accept | user488802 | ||
| Jun 21, 2023 at 22:27 | |||||
| Jun 21, 2023 at 20:46 | history | edited | Dave Benson | CC BY-SA 4.0 |
deleted 46 characters in body
|
| Jun 21, 2023 at 13:15 | history | edited | Dave Benson | CC BY-SA 4.0 |
added 545 characters in body
|
| Jun 21, 2023 at 11:06 | history | edited | Dave Benson | CC BY-SA 4.0 |
added 185 characters in body
|
| Jun 21, 2023 at 9:19 | vote | accept | user488802 | ||
| Jun 21, 2023 at 22:05 | |||||
| Jun 21, 2023 at 9:11 | history | edited | Dave Benson | CC BY-SA 4.0 |
added 136 characters in body
|
| Jun 21, 2023 at 8:54 | history | answered | Dave Benson | CC BY-SA 4.0 |