Timeline for Finite groups with elements of order n
Current License: CC BY-SA 2.5
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 11, 2015 at 22:01 | answer | added | Marty Isaacs | timeline score: 7 | |
| Sep 27, 2015 at 10:12 | answer | added | Shahrooz | timeline score: 2 | |
| Oct 28, 2014 at 1:25 | comment | added | Doug | The example of the Heisenberg group doesn't necessarily satisfy all elements having the same order. For example, the Heisenberg group over $C_2$ is $ D_8$. | |
| Jul 15, 2010 at 5:34 | vote | accept | falagar | ||
| Jul 14, 2010 at 11:20 | answer | added | Francesco Polizzi | timeline score: 3 | |
| Jul 14, 2010 at 10:03 | answer | added | Bugs Bunny | timeline score: 10 | |
| Jul 14, 2010 at 6:25 | answer | added | Victor Protsak | timeline score: 18 | |
| Jul 14, 2010 at 5:59 | comment | added | falagar | Well, I understand now that $n$ must be prime and group might be nonabelian. So there are no straitforward generalizations of case $n=2$. Originally, I was looking for the statement like '... then the group is isomorphic to such or such group'. Now I think there are no such classification and the question is closed. | |
| Jul 14, 2010 at 5:47 | comment | added | Pete L. Clark | @falagar: Could you please address what remains of your question in light of my comment above? "Could it be somehow generalized" is not a good question: virtually anything can be somehow generalized. What exactly are you looking for? | |
| Jul 14, 2010 at 5:45 | history | edited | falagar | CC BY-SA 2.5 |
corrected statement
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| Jul 14, 2010 at 5:35 | comment | added | Pete L. Clark | If all elements have the same order, then since the identity has order $1$, the group must be trivial. So surely you mean to exclude the identity. Also, it is easy to see that if all non-identity elements have the same order, that order must be a prime number $p$. If $p>2$, then it is well-known that there are finite groups of exponent $p$ which are not commutative: e.g. the group of upper triangular $3 \times 3$ matrices over $\mathbb{F}_p$ with $1$'s on the main diagonal ("finite Heisenberg group"). | |
| Jul 14, 2010 at 5:35 | comment | added | Yemon Choi | Can I also ask if you were looking at this question from curiosity, or if it was suggested to you as something to think about? | |
| Jul 14, 2010 at 5:33 | comment | added | Steve D | $n$ must be a prime. | |
| Jul 14, 2010 at 5:29 | comment | added | Daniel Barter | if n=p and the group is abelian, you can prove that it is a $\mathbb{Z} / p \mathbb{Z}$ vector space | |
| Jul 14, 2010 at 5:24 | history | asked | falagar | CC BY-SA 2.5 |