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About the Product of Two Commuting Elements

  1. Sep 20, 2016 #1
    1. The problem statement, all variables and given/known data
    I am asked to offer an example of two commuting elements whose product does not have an order equal to the least common multiple of their individual orders.

    2. Relevant equations


    3. The attempt at a solution
    Consider ##-1## and ##1## in ##\mathbb{Z}##. Then ##1+(-1) = 0## which has an order of ##1##, but the order of ##-1## and ##1## is infinity.

    Would this be an acceptable answer? I find it unsettling for some reason, but I cannot see anything wrong with it.
     
  2. jcsd
  3. Sep 20, 2016 #2

    fresh_42

    Staff: Mentor

    This looks a bit like cheating. What is the least common multiple of ##\infty## and ##\infty##?
    You should consider ##( \mathbb{Z_{12}}\, , \,+) = (\mathbb{Z}/12 \mathbb{Z} \, , \, +)## instead, i.e. the remainders from division by ##12## or simply the little hand on the clock.
     
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