On Thursday, 19 July 2012 19:33:48 UTC+8, Javier López Peña wrote: > > 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. >
when GAP works with these kinds of groups (any soluble group, in fact), it treats them as power-commutator presented groups. This allows for relatively efficient procedures (many algorithmic problems, which are unsolvable for general finitely presented groups, and solvable for power-commutator presented groups) > >> 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. > I didn't mean to use "natural" in categorical sense here. > > 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. > sure, you'd get PSL as a permutation group. By you can get SL as a matrix group, no problem about it. > > 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. > just use matrix groups rather than projective groups... > > > >> 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. > Huh? I don't think you are right here. Dima > I would like to have an option to see them as matrix groups. > > Cheers > J > > -- -- To post to this group, send an email to [email protected] To unsubscribe from this group, send an email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
