Dear GAP-Forum: I also found the Cycle Index and Polya Polynomial of the Group generated by all the primes coprime to 24 (Totient). (1,5,7,11,13,17,19,23) Here are my results. My question is, what is the Group-theoretical name for this? (Something like D8.2 X S3, having order 192?) G:= Group( [ (1,2,3,4,5,6,7,8), (1,3)(2,6)(5,7), (1,5)(3,7), (1,7)(2,6)(3,5), ( 9,10), ( 9,10,11) ] ); (Convert to Cartesian...) Cycle Index: 1/192*x_1^24+1/96*x_1^12*x_2^6+1/64*x_1^8*x_2^8+1/24*x_1^6*x_2^9+1/32*x_1^4*x_\2^10+1/8*x_1^2*x_2^11+5/48*x_2^12+1/96*x_3^8+1/48*x_3^4*x_6^2+1/6*x_4^6+1/12*x\_3^2*x_6^3+5/96*x_6^4+1/6*x_8^3+1/12*x_12^2+1/12*x_24 Polya: x_1^24+x_1^23+7*x_1^22+23*x_1^21+97*x_1^20+294*x_1^19+870*x_1^18+2051*x_1^17+4\272*x_1^16+7352*x_1^15+10980*x_1^14+13790*x_1^13+15008*x_1^12+13790*x_1^11+109\80*x_1^10+7352*x_1^9+4272*x_1^8+2051*x_1^7+870*x_1^6+294*x_1^5+97*x_1^4+23*x_1\^3+7*x_1^2+x_1+1 Thanks PGH
To: [EMAIL PROTECTED]: [EMAIL PROTECTED]; [EMAIL PROTECTED]: Re: [GAP Forum] Cartesian Group Direct ProductFrom: [EMAIL PROTECTED]: Sun, 6 Jan 2008 16:06:22 -0600Dear Alexander Hulpke, Dear GAP-Forum, Works great. The relabelling was confusing, but that is because the permutation for S3 permutes the origin, so to speak, using (5,6), but it doesn't matter, I get the correct CycleIndex and expanding it out is easy, now that I know what to do. This is the Polya Polynomials I get for D8 X S3 and C8 X C3 respectively, I believe they are correct: x^24+x^23+9*x^22+31*x^21+151*x^20+496*x^19+1553*x^18+3777*x^17+8028*x^16+13981\ *x^15+21042*x^14+26516*x^13+28898*x^12+26516*x^11+21042*x^10+13981*x^9+8028*x^\ 8+3777*x^7+1553*x^6+496*x^5+151*x^4+31*x^3+9*x^2+x+1 x^24+x^23+12*x^22+85*x^21+446*x^20+1771*x^19+5620*x^18+14421*x^17+30667*x^16+5\ 4484*x^15+81752*x^14+104006*x^13+112720*x^12+104006*x^11+81752*x^10+54484*x^9+\ 30667*x^8+14421*x^7+5620*x^6+1771*x^5+446*x^4+85*x^3+12*x^2+x+1 Paul HjelmstadITS Enterprise Security763-582-6547 (x46547)[EMAIL PROTECTED] 'PAUL HJELMSTAD' <[EMAIL PROTECTED]> 01/05/2008 12:05 PM To [EMAIL PROTECTED] cc [EMAIL PROTECTED] Subject Re: [GAP Forum] Cartesian Group Direct Product Dear Alexander Hulpke, Dear GAP-Forum,Thanks! I realized after I sent my last message, that I was not understanding the meaning of 'Cartesian' correctly, but I see you understood what I meant. Someone else called what I am looking for the 'cross-product' representation, or what you call the transitive product representation.Finding the CycleIndex should be relatively easy here, but expanding to the Polya Polynomialhas proved difficult, merely because of a problem with variables. ThanksPGH>From: Alexander Hulpke <[EMAIL PROTECTED]>>To: PAUL HJELMSTAD <[EMAIL PROTECTED]>>CC: GAP Forum <[EMAIL PROTECTED]>>Subject: Re: [GAP Forum] Cartesian Group Direct Product>Date: Fri, 4 Jan 2008 12:35:13 -0700>>Dear Paul Hjelmsted, Dear GAP-Forum,>>>The problem with this is that it merely seems to shuffle between the >>non-Cartesian>>form of the permutations, that is, for example, sending (123) to (567) >>(Second Embedding) or merely leaving at at (123) (First Embedding) but I >>may be doing something wrong.>>>>I am not getting anything Cartesian-wise. Perhaps I must leave D4 >>(actually called Dihedral(8)),>>as a pc-group and not a perm group?>>>>After I get this right, I need to generate the CycleIndex, and then >>expand it in a manner you indicated, to get the full Polya Polynomial, >>whose coefficients will be useful to me (especially to find how many >>octads there are under D8 X S3 (Dihedral(16) X Symmetric(3)) and other >>issues>>>>I need these generators for D4 X S3:>>>>(0,3,6,9)(1,4,7,10)(2,5,8,11)>>(0,4,8)(1,5,9)(2,6,10)(3,7,11)>>(1,7)(3,9)(5,11)>>(1,11)(2,10)(3,9)(4,8)(5,7)>>>>Or something with the same meaning>>OK. You want a different representation for the direct product. GAP gives >you by default the intransitive action (which has smaller degree), you >would prefer the transitive product action.>>The easiest way to construct this group is to let the intransitive direct >product act on the cartesian product of the domains:>>gap> d4:=DihedralGroup(IsPermGroup,8);>Group([ (1,2,3,4), (2,4) ])>gap> s3:=SymmetricGroup(3);>Sym( [ 1 .. 3 ] )>gap> d:=DirectProduct(d4,s3);>Group([ (1,2,3,4), (2,4), (5,6,7), (5,6) ])>gap> cart:=Cartesian([1..4],[5..7]);>[ [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], [ 2, 5 ], [ 2, 6 ], [ 2, 7 ], [ 3, 5 ],> [ 3, 6 ], [ 3, 7 ], [ 4, 5 ], [ 4, 6 ], [ 4, 7 ] ]>gap> hom:=ActionHomomorphism(d,cart,OnTuples,'surjective');><action epimorphism>>gap> prod:=Image(hom);>Group([ (1,4,7,10)(2,5,8,11)(3,6,9,12), (4,10)(5,11)(6,12),> (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,2)(4,5)(7,8)(10,11) ])>>>Up to labelling (which is due to the arrangement of the pairs in `cart') >these are the generators you listed. You can use the map `hom' to go back >to the intransitive direct product and use its decomposition functions.>>>Best wishes,>> Alexander Hulpke>>-- Colorado State University, Department of Mathematics,>Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA>email: [EMAIL PROTECTED], Phone: ++1-970-4914288>http://www.math.colostate.edu/~hulpke>> CONFIDENTIALITY NOTICE: The information in this message, and any files transmitted with it, is confidential, may be legally privileged, and intended only for the use of the individual(s) named above. Be aware that the use of any confidential or personal information may be restricted by state and federal privacy laws. If you are not the intended recipient, do not further disseminate this message. If this message was received in error, please notify the sender and delete it. _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
