Re: Question on factor group calculations
- To: mathgroup at smc.vnet.net
- Subject: [mg38370] Re: Question on factor group calculations
- From: "Diana" <diana53xiii at earthlink.remove13.net>
- Date: Fri, 13 Dec 2002 04:09:16 -0500 (EST)
- References: <at4bvq$en5$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Andrzej, This is great! I can't wait to play with this tonight. I have printed up your e-mails, and have added them to my Abstract Algebra notes for this semester. My teacher is going to get a kick out of seeing how this one is done. Thank you, Diana Mecum "Diana" <diana53xiii at earthlink.remove13.net> wrote in message news:at4bvq$en5$1 at smc.vnet.net... > Math friends, > > I am trying to create a multiplication table for the factor groups, > > (Z_4 (+) Z_12)/<(2,2)> > > I understand how to list the elements of (Z_4 (+) Z_12). This is done with: > > Z4Z12 = Flatten[Outer[List, Range[0, 3], Range[0, 11]], 1] > > I would like to be able to figure out how to list the eight factor groups > with a calculation with (2,2). This would be in modulo 4,12 arithmetic. > > As a workaround, I defined (Z_4 (+) Z_12)/<(2,2)> as the eight cosets > defined below: > > multZ4Z12[{a_, b_}, {c_, d_}] := {Mod[a + c, 4], Mod[b + d, 12]} > > Multiplication[Z4Z12, multZ4Z12] // TableForm; > > Coset1 = {Z4Z12[[1]], Z4Z12[[27]], Z4Z12[[5]], Z4Z12[[31]], Z4Z12[[9]], > Z4Z12[[35]]} > > Coset2 = {Z4Z12[[2]], Z4Z12[[28]], Z4Z12[[6]], Z4Z12[[32]], Z4Z12[[10]], > Z4Z12[[36]]} > > Coset3 = {Z4Z12[[3]], Z4Z12[[29]], Z4Z12[[7]], Z4Z12[[33]], Z4Z12[[11]], > Z4Z12[[25]]} > > Coset4 = {Z4Z12[[4]], Z4Z12[[30]], Z4Z12[[8]], Z4Z12[[34]], Z4Z12[[12]], > Z4Z12[[26]]} > > Coset5 = {Z4Z12[[37]], Z4Z12[[15]], Z4Z12[[41]], Z4Z12[[19]], Z4Z12[[45]], > Z4Z12[[23]]} > > Coset6 = {Z4Z12[[38]], Z4Z12[[16]], Z4Z12[[42]], Z4Z12[[20]], Z4Z12[[46]], > Z4Z12[[24]]} > > Coset7 = {Z4Z12[[39]], Z4Z12[[17]], Z4Z12[[43]], Z4Z12[[21]], Z4Z12[[47]], > Z4Z12[[13]]} > > Coset8 = {Z4Z12[[40]], Z4Z12[[18]], Z4Z12[[44]], Z4Z12[[22]], Z4Z12[[48]], > Z4Z12[[14]]} > > I was not able to figure a way to create a multiplication table with these > eight elements, because of the multiple part modulo addition. > > There must be a way to multiply <(2,2)> by different elements of the > external direct product, and a way to compute the multiplication table of > the factor groups. Can someone help? > > Thanks, > > Diana > > ===================================================== > "God made the integers, all else is the work of man." > L. Kronecker, Jahresber. DMV 2, S. 19. > > >