On Thursday, 19 July 2012 15:37:32 UTC+8, Javier López Peña wrote: > > I understand that from some point of view mixing groups and > their representations is a bad idea, but many groups are naturally > defined as transformation groups and using a matrix presentation > is just as natural as describing them by permutations, or even more so. > Not to mention the huge size some of the "constructions by > permutations" have. > > let me nitpick first by saying that in group theory "presentation" means "presentation by generators and relations" whereas you mean a (linear) "representation".
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>. But of course this requires quite a bit of machinery to be useful. Z_n can also be naturally represented as a permutation group, with <(1,2,...,n)> the most straightforward one. Already what GAP is doing is essentially constructing the permutation matrix from (1,2,...,n). > For groups with a canonical matrix form I'd vote for the mythical > python construction. Surely everybody agrees PSL(2,16) > has a "natural" presentation by 2x2 matrices over GF(16), and > there is no reason we shouldn't be able to use it. > These are typically available in GAP (and this in Sage) already. GAP has all these GL, SL, SU, Sp, etc. e.g: sage: gg=SU(5,2) sage: print gg Special Unitary Group of degree 5 over Finite Field of size 2 sage: gg.order() 13685760 Dima -- -- 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
