Dear Alireza, On 25 Jul 2014, at 12:46, arashr...@kashanu.ac.ir wrote:
> > Dear GAP Forum Members, > > Suppose G is a group of a given order n and m is a divisor of n. I am > interesting to the problem of computing all subgroups of G of order n. > Please notice that if n is enough large, it is not efficient to first > compute all subgroups of G and then obtain subgroups of order m. For > example I need to the subgroups of order 32 in a group of order 1024. > Any suggestions or comments will be highly appreciated. > > Regards, Alireza Here there is a collection of links that may be useful: 7.7: How do I get the subgroups of my group? http://gap-system.org/Faq/faq.html#7.7 39.21 Specific Methods for Subgroup Lattice Computations http://gap-system.org/Manuals/doc/ref/chap39.html#X85E613D57F28AEFF In particular, for the case of a solvable group there is SubgroupsSolvableGroup with the optional argument which permits to specify the order of the group: gap> G:=DihedralGroup(1024); <pc group of size 1024 with 10 generators> gap> l:=SubgroupsSolvableGroup(G,rec(consider:=ExactSizeConsiderFunction(32))); [ Group([ ]), Group([ f10 ]), Group([ f10, f9 ]), Group([ f10, f9, f8 ]), Group([ f10, f9, f8, f7 ]), Group([ f10, f9, f8, f7, f6 ]), Group([ f10, f9, f8, f7, f6, f5 ]), Group([ f10, f9, f8, f7, f6, f5, f4 ]), Group([ f10, f9, f8, f7, f6, f5, f4, f3 ]), Group([ f10, f9, f8, f7, f6, f5, f4, f3, f1 ]), Group([ f10, f9, f8, f7, f6, f5, f4, f1 ]), Group([ f10, f9, f8, f7, f6, f5, f1 ]), Group([ f10, f9, f8, f7, f6, f1 ]), Group([ f10, f9, f8, f7, f1 ]), Group([ f10, f9, f8, f7, f6, f5, f4, f3, f2 ]), Group([ f10, f9, f8, f7, f6, f5, f4, f3, f1*f2 ]), Group([ f10, f9, f8, f7, f6, f5, f4, f1*f2 ]), Group([ f10, f9, f8, f7, f6, f5, f1*f2 ]), Group([ f10, f9, f8, f7, f6, f1*f2 ]), Group([ f10, f9, f8, f7, f1*f2 ]), Group([ f10, f9, f8, f7, f6, f5, f4, f3, f1, f2 ]) ] gap> List(l,Size); [ 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 256, 128, 64, 32, 512, 512, 256, 128, 64, 32, 1024 ] This list is redundant by reasons explained below, so we have to filter out groups of order 32 from this list: gap> Filtered(l, g -> Size(g)=32); [ Group([ f10, f9, f8, f7, f6 ]), Group([ f10, f9, f8, f7, f1 ]), Group([ f10, f9, f8, f7, f1*f2 ]) ] - these are representatives of conjugacy classes of subgroups of order 32. SubgroupsSolvableGroup implements the algorithm published in [Hulpke, A., Computing subgroups invariant under a set of automorphisms, J. Symbolic Comput., 27 (4) (1999), 415–427] and computes subgroups of a solvable group G, using the homomorphism principle. It returns a list of representatives up to G-conjugacy. Just quoting the manual, "the 'consider' function does not provide a perfect filter. It is guaranteed that all subgroups fulfilling the condition are returned, but not all subgroups returned necessarily fulfill the condition." Anyhow, this is a little bit faster than computing all subgroups for the group from the example (and may be much more faster for other groups): gap> G:=DihedralGroup(1024); <pc group of size 1024 with 10 generators> gap> l:=SubgroupsSolvableGroup(G,rec(consider:=ExactSizeConsiderFunction(32)));;time; 164 gap> G:=DihedralGroup(1024); <pc group of size 1024 with 10 generators> gap> l:=SubgroupsSolvableGroup(G);;time; 226 gap> Hope this helps Alexander P.S. Perhaps in an interactive session I would be typing just the following: gap> G:=DihedralGroup(1024); <pc group of size 1024 with 10 generators> gap> cc:=ConjugacyClassesSubgroups(G);; gap> Filtered(cc, c -> Size(Representative(c))=32); [ Group( [ f10, f9, f8, f7, f6 ] )^G, Group( [ f10, f9, f8, f7, f1*f2 ] )^G, Group( [ f10, f9, f8, f7, f1 ] )^G ] gap> _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
