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External direct product

  1. Aug 11, 2011 #1
    1. The problem statement, all variables and given/known data

    Explain why external direct products z8 + z4 and z80000000+ z4000000 have same number of elements of order 4?

    2. Relevant equations



    3. The attempt at a solution

    Z 8 = { 0, 1, 2, 3, 5, 6, 7, } , order of elements : 0 =1, 1=8, 2=4 , 3=8, 4=2, 5=8, 6=4, 7=8.

    Z 4= { 0, 1, 2, 3} order of elements : 0= 1, 1=4, 2=2, 3=4.

    Hence, in the EDP Z8 + Z4...it seems that there are 12 elements of order 4

    (0, 1), (0, 3), (2,0) (2, 1), (2,2) (2, 3) (4,1) (4,3) (6,0) (6,1) (6,2) (6, 3).

    Please explain how to count the no. of elements of order 4 in z80000000+ z4000000
     
  2. jcsd
  3. Aug 12, 2011 #2

    micromass

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    Can you use the fundamental theorem of abelian groups to write [itex]\mathbb{Z}_{80000000}[/itex] and [itex]\mathbb{Z}_{4000000}[/itex] differently??
     
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