Timeline for Characterization of Frobenius complements
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Sep 5, 2020 at 7:09 | answer | added | Qiaochu Yuan | timeline score: 3 | |
| Jan 6, 2019 at 17:40 | answer | added | Hempelicious | timeline score: 6 | |
| Jan 8, 2015 at 23:27 | comment | added | Geoff Robinson | @FriederLadisch : It's good to have an explicit reference for at least one direction. | |
| Jan 8, 2015 at 16:20 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
added 409 characters in body
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| Jan 8, 2015 at 16:11 | comment | added | Frieder Ladisch | Passman's book contains as Theorem 18.1.v. the statement that a Frobenius complement $G$ has a complex representation such that each non-identity element has no fixed points on the representation space. (The proof is as in Geoff Robinson's answer.) I don't know a reference for the other direction (but the proof is similar, as Geoff mentions). I looked into Huppert's "Endliche Gruppen", but I wasn't able to find the statement there. | |
| Jan 7, 2015 at 21:37 | comment | added | LSpice | Thanks, and no worries for the delay! | |
| Jan 7, 2015 at 16:39 | comment | added | Joonas Ilmavirta | @LSpice, the thing I'm writing about is related to this earlier post: mathoverflow.net/q/184606/55893 See the second motivation paragraph. The intended audience is mainly the inverse problems community. (Sorry for taking a week to write a simple comment.) | |
| Jan 1, 2015 at 18:06 | comment | added | LSpice | The description of the result and of your intended audience is intriguing. Would you be comfortable saying anything about a 'big picture' of what you are writing? | |
| Jan 1, 2015 at 12:22 | comment | added | Henri Johnston | If you just want the statement, then you can use "Profinite Groups" by Luis Ribes and Pavel Zalesskii - see Theorem 4.6.1. For the proof, they refer to Huppert's book in my comment above. | |
| Jan 1, 2015 at 12:19 | comment | added | Henri Johnston | I don't have the book to hand, so I can't be sure, but I think it could be in "Endliche Gruppen I" by Huppert, 1967. Section V.8 covers Frobenius groups. | |
| Dec 30, 2014 at 21:27 | answer | added | Geoff Robinson | timeline score: 9 | |
| Dec 30, 2014 at 17:27 | comment | added | Derek Holt | @GeoffRobinson Yes, good point! | |
| Dec 30, 2014 at 16:55 | comment | added | Geoff Robinson | @Derek Holt: You don't actually need Thompson's Theorem. Frobenius's original theorem, together with basic facts about coprime action are enough. If $G = KH$ is a Frobenius group with kernel $K$ and complement $H,$ then ${\rm gcd}(|K|,|H|) = 1$ and for each prime divisor $q$ of $|K|$, $K$ has an H-invariant Sylow $q$-subgroup ( on which the action of $H$ is still faithful and "Frobenius"). Hence we can find a Frobenius subgroup of the form $VH$ where $V$ is an elementary Abeliaan $q$-group on which $H$ acts irreducibly. | |
| Dec 30, 2014 at 14:10 | comment | added | Derek Holt | Since (finite) Frobenius kernels are nilpotent (a result of Thompson), a minimal normal subgroup of a Frobenius group is elementary abelian, and so it is not hard to see that a finite group is a Frobenius complement if and only if it has a fixed point free representation over a finite field in coprime characteristic. I would guess that to get the equivalence with fixed point free complex representations, you would need to know about the equivalence between complex representations and representations in finite coprime characteristic. I understand why you prefer a direct reference! | |
| Dec 30, 2014 at 11:07 | history | asked | Joonas Ilmavirta | CC BY-SA 3.0 |