On Thursday, July 19, 2012 9:52:26 AM UTC+1, Dima Pasechnik wrote: > > let me nitpick first by saying that in group theory > "presentation" means "presentation by generators and > relations" whereas you mean a (linear) "representation". >
Fine, maybe I should have use "realization" or "imploementation" instead. I didn't want to mix up "representation" in the group theory sense and in the "internal representation as a sage object" sense. > In this way of thinking, the most compact way to represent Z_n is by > generators and relations, i.e. Z_n=<a| a^n=1>. > Agreed. There is a patch (#12339) bringing GAP's machinery for finitely presented groups to Sage, and I believe this should be the default implementation for groups defined by generators and relations unless there are severe performance issues, in which case an easy way to opt to a more efficient implementation should be provided. > Z_n can also be naturally represented as a permutation group, with > <(1,2,...,n)> the most straightforward one. > Here I don't agree. There is nothing "natural" in the mathematical sense (functorial, since you wanted to nitpick) in that representation. The closest thing is to consider the Cayley representation of a (finite) group acting on itself, but this is terribly inefficient and does not respect group homomorphisms. Many groups are internally built in GAP/Sage by means of some "minimal" permutation representation even if they "naturally" are defined as (quotients of) matrix groups, try to define PSL(2,7) and look at the generators to see what I mean. I am not aware of an easy way of obtaining a sage version of those groups that works with matrices; since I am currently performing some heavy computations involving simple groups of up to order 100000 I would certainly benefit from a faster/more efficient internal implementation than the default one as groups of permutations. > These are typically available in GAP (and this in Sage) already. > GAP has all these GL, SL, SU, Sp, etc. e.g: > Full matrix groups are, simple groups of Lie type are internally constructed as permutation groups. I would like to have an option to see them as matrix groups. Cheers J -- -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
