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Finding the order

  1. Oct 31, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the order of the element 5 + <3> in the quotient group II18/ <3>.


    2. Relevant equations
    II18 = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
    <3> = {3,6,9,12,15,18}
    5 + <3> = {5,8,11,14,17,2}
    order is the number of elements

    3. The attempt at a solution
    I am thinking this II18/ <3> equals {3,6,9,12,15,18} and I have to find the order of that plus 5.

    So I got the order as 6.
     
    Last edited: Oct 31, 2009
  2. jcsd
  3. Nov 1, 2009 #2
    I assume II18 means Z18, i.e. integers mod 18 under addition mod 18.


    OK, and 18=0=the identity in II18.

    Not in this case. Order of a group is the number of elements. Order of an element is the smallest positive integer n such that a^n=aa...a=identity (in multiplicative notation), i.e. such that na=a+a+...+a=identity (in additive notation, which applies to this example). You have to find the order of the element 5 + <3> in the group II18/<3>



    This part is all wrong. II18/ <3> equals {<3>,1+<3>,2+<3>}. The identity in this group is <3>, or maybe you call it 18+<3>.

    Add the element 5+<3> to itself repeatedly until you get the identity <3>. Note that 5+<3> = 2+<3>. How many times did you add it to itself? That is, what was the smallest positive integer n such that n times 5+<3> equals the identity <3>? That n is your answer.
     
  4. Nov 1, 2009 #3
    so I did that and I came up with:
    0+<3> = {0,3,6,9,12,15}
    1+<3> = {1,4,7,10,13,16}
    2+<3> = {2,5,8,11,14,17}
    3+<3> = {3,6,9,12,15,0}
    4+<3> = {4.7.10.13.16.1}
    5+<3> = {5,8,11,14,17,2}
    So I get that 0 & 3 are tha same, 1&4 are the same and 2&5 are the same.
    So I have to do n(5+<3>)= {3,6,9,12,15,0}? b/c nothin times 2 will give me 15.
     
    Last edited: Nov 1, 2009
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