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Element order in factor groups

  1. Oct 28, 2008 #1
    I'm currently studying factor groups in abstract algebra and needed some help understanding how to determine the order of an element in a factor group

    Suppose I have Z (mod 12) / <4>. And I choose some random element from Z (mod 12) such as 5 or 7. How would I go about determining the order of 5 + <4> or 7 + <4> in the factor group. The way I understand is that in Z (mod 12) / <4> we collapse everything in <4> to the identity in Z (mod 12) so that 5 for instance would have order |{5,10,3,8}| = 4 because 8 is an identity in the factor group. Any help or correction on my admittedly poor understanding would be great.
     
  2. jcsd
  3. Oct 28, 2008 #2
    From a theorem, i believe, if the index of a normal group, N, in a group G is say, like in your case 4. then the order of the factro group G/N is 4. So, by lagranges theorem for every el in G/N, say C, C^4=N (identity). Now in your case, the order of Z_12/<4> is 4, so there will be 4 distinct cosets of <4> in Z_12, all of length 3, which makes me think, that the order of any of the el. in Z_12/<4> is going to be 3.
     
  4. Nov 2, 2010 #3
    The way I understand it is list the elements of the factor group. <4>={0,4,8} then list the factors of the element you are trying to find the order of. 5, 10, 3, 8 Stop it took 4 times to get to an element in the factor group. Therefore the element has an order of 4.
     
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