Timeline for In which fixed-point free representations is the sum of every 3 elements invertible?
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15 events
| when toggle format | what | by | license | comment | |
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| Apr 1, 2014 at 11:29 | comment | added | Geoff Robinson | The paper arxiv.org/pdf/math/0005037.pdf is illuminating, though you are looking at a very special case | |
| Mar 31, 2014 at 21:21 | comment | added | Dima Pasechnik | OK, Geoff, I must say I was totally humbled by your reference to Clifford's theorem :-) | |
| Mar 31, 2014 at 11:11 | comment | added | Geoff Robinson | $F(G)$ is the largest nilpotent normal subgroup of $G$. A finite group $G$ can be a fixed point free group if and only if it can occur as a Frobenius complement in some Frobenius group. A Froobenius complement has cyclc Sylow $q$-subgroups for all odd primes $q$. It is reasonably well known that the only non solvable Frobenius complement is ${\rm SL}(2,5).$ | |
| Mar 31, 2014 at 11:06 | comment | added | A.B. | Thanks! Just 2 quick questions: 1. What does F(G) stand for? 2. Is an odd fixed-point free group and a Frobenius complement the same thing? (sorry if these are very basic questions, but I didn't learn these concepts) | |
| Mar 31, 2014 at 11:06 | vote | accept | A.B. | ||
| Mar 31, 2014 at 10:59 | comment | added | Geoff Robinson | I believe that is the case | |
| Mar 31, 2014 at 10:57 | comment | added | A.B. | So basically, if I work with odd fixed-point free groups then I'm safe? | |
| Mar 31, 2014 at 9:12 | history | edited | Geoff Robinson | CC BY-SA 3.0 |
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| Mar 31, 2014 at 8:55 | comment | added | Geoff Robinson | @Dima Pasechnik: Sorry, I misread the question- for some reason, I thought that the question was about $3$-dimensional representations. | |
| Mar 31, 2014 at 8:52 | history | edited | Geoff Robinson | CC BY-SA 3.0 |
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| Mar 31, 2014 at 7:39 | history | edited | Geoff Robinson | CC BY-SA 3.0 |
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| Mar 31, 2014 at 5:13 | comment | added | A.B. | I have to repeat @DimaPasechnik 's question - why does there exist an irreducible representation of dimension 3? I might take the quaternions for example (they do not interest me since this is an even group, but SmallGroup(275,1) on GAP also seems problematic). Another thing: notice that I allow for the sum to be 0, just not a non-zero singular matrix. | |
| Mar 30, 2014 at 20:20 | comment | added | Dima Pasechnik | Why would 3 even divide $|G|$ ? And why must the normal 3-complement be Abelian? | |
| Mar 30, 2014 at 20:16 | history | edited | Geoff Robinson | CC BY-SA 3.0 |
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| Mar 30, 2014 at 20:10 | history | answered | Geoff Robinson | CC BY-SA 3.0 |