Timeline for Why do we associate a graph to a ring?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 18, 2020 at 17:10 | comment | added | Sh.M1972 | I would like to mention a similar situation in group theory: We can associate a Lie ring to a group. OK, this not only a curiosity! After studying the basic properties of this correspondence, we use it to solve very hard problems of group theory, like, Burnside's restricted problem, fixed points of automorphisms, and many many deep results of group theory. This correspondence, is very useful, it carries results from Lie theory to group theory. So it is not only a story of the form "if the group has property P, the the Lie ring has property Q". Math, does not work so. | |
| Oct 18, 2020 at 16:59 | comment | added | Sh.M1972 | .... " in which classes of groups, knowing the graph gives us good information of the group?" For example, the paper which is addressed by Benjamin Steinberg is a very good paper about linear groups, but it does not use commuting graph to solve a group problem, it is about the "diameter of the commuting graph of some linear groups", a problem of "NEW KIND". | |
| Oct 18, 2020 at 16:56 | comment | added | Sh.M1972 | If you review all papers written in this area, you will realize that no deep result of graph theory is applied, only definition of a graph and elementary notions like diameter, coloring number, planarity, ... are considered. In think the main question of OP is this: is there any hard problem of group theory that is solved after introducing commuting or non-commuting graphs? I think, the answer is no. BUT, there are new problems, and new questions of the form "for which kind of groups the graph has a property Q? | |
| Oct 18, 2020 at 16:50 | comment | added | Sh.M1972 | Let me give a simple example: suppose you say the diameter of the non-commuting graph of a group $G$ is at least 4. We can restate this graph theory sentence in the language of groups: There are for elements $a$, $b$, $c$, $d$ in $G$ such that any pair of them are non-commuting. Every result about the graphs associating to groups can be translated in this form. They only have two benefits: the group theoretic property becomes easy to understand and imagination. Also we can ask NEW questions about group-graph properties (if group has the property P, then the graph has the property Q) . | |
| Oct 18, 2020 at 16:45 | comment | added | Sh.M1972 | I don't have the book of Suzuki right now, so would you please send me a link. I am sure that it has no relation to objects like commuting or non-commuting graphs, and as i said, with a large probability, that Fischer graph is only a device to simplify group theoretic long and complicated sentences. I mean, every thing down by introducing a graph in group theory, also can be done without graph, in purely group theoretic language, too. Unfortunately, there is no evidence of a serious application of graph theory via commuting or non-commuting graph in group theory. | |
| S Oct 18, 2020 at 12:38 | history | answered | Alireza Abdollahi | CC BY-SA 4.0 | |
| S Oct 18, 2020 at 12:38 | history | made wiki | Post Made Community Wiki by Alireza Abdollahi |