Timeline for Burnside group $B(2, 3)$ has $27$ elements, isomorphic to unitringular matrix group $\text{UT}(3, 3)$?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 4, 2016 at 13:48 | comment | added | Geoff Robinson | The notes of G. Traustason, which can be found at people.bath.ac.uk/gt223/paper27.pdf , make it clear that Burnside knew that the Burnside Problem was true for groups of exponent $3$, and that any two conjugate elements commute in a group of exponent $3$, from which it follows directly that a two-generator group of exponent $3$ has a commutator group which is central and cyclic. So I am now persuaded that he was aware of the structure of $B(2,3)$ in particular. | |
| Oct 1, 2016 at 15:16 | comment | added | David A. Jackson | I am somewhat surprised that no one has mentioned Chapter 18 of Marshall Hall's book, The Theory of Groups. He does cite the Levi-van der Waerden paper and proves that $B(r,3)$ has order $3^{m(r)}$ where $m(r) = r$ $ +$ ${r}\choose 2 $ + ${r}\choose 3 $. Hall also includes information about the commutator power structure for elements of $B(r,3)$. | |
| Oct 1, 2016 at 11:43 | comment | added | Geoff Robinson | Well, sorry, yes, it was an unreasonable question since no-one could be sure of such a thing at this point. I too am unsure in the other direction. | |
| Oct 1, 2016 at 10:08 | comment | added | Igor Rivin | @GeoffRobinson I am not sure, but this article: www-groups.dcs.st-and.ac.uk/history/HistTopics/… seems to imply that he did not. | |
| Oct 1, 2016 at 9:30 | comment | added | Geoff Robinson | Are you sure Burnside didn't know this fact himself? | |
| Oct 1, 2016 at 8:38 | history | answered | Igor Rivin | CC BY-SA 3.0 |