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Group order from a presentation

  1. Dec 17, 2012 #1
    Hello.

    I have been looking at some questions from old exams that I am preparing for, and I have some trouble with the kind of problems that I will now give an example of.

    1. The problem statement, all variables and given/known data

    Let [itex]G = (a,b,c | a^4 = 1, b^2 = a^2, bab^{-1} = a^{-1}, c^3 = 1, cac^{-1} = b, cbc^{-1} = ab)[/itex]. Determine the order of this group.

    2. Relevant equations
    null

    3. The attempt at a solution
    The relations imply that one can move c to the left past all a,b. Also, b can be moved to the left past all a, and hence we can express all elements of this group as products of a power of a, power of b and power of c (in that order). There are at most 24 elements. Now, I have not found any relations that imply orders of a and c are less than 4 and 3, so I assume the group has order 24, but how can I prove this? Easiest would be to find a group generated by 3 (or less) elements that satisfy these properties, and map G surjectively into it, but I cannot expect to memorize all low order groups.

    On groups with 2 generators and simpler relations, I usually just do mind-numbing computation of the left regular representation of this group presentation. Then if the relations do not collapse the group, I can map it surjectively into the permutation group I obtained and draw conclusions about order. What can I do in this case, when there are 3 generators and relations pretty much complicated enough so that working out a regular representation on paper is out of question. Are there any popular tricks?
     
  2. jcsd
  3. Dec 18, 2012 #2
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