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Question on factor group calculations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg38249] Question on factor group calculations
  • From: "Diana" <diana53xiii at earthlink.remove13.net>
  • Date: Tue, 10 Dec 2002 04:10:03 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

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.



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