Find all possible orders of elements in Sym(5)

Find all numbers n such that S_5 contains an element of order n.


Note that the only possible cycle decompositions for elements in S_5 (up to the order of the cycles) are 1, (- -), (- - -), (- -)(- -), (- -)(- - -), (- - - -), and (- - - - -). By a previous theorem, the order of a product of disjoint cycles is the least common multiple of the lengths of the factors. Thus, the possible orders for elements in S_5 are (in increasing order) 1, 2, 3, 4, 5, and 6.

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Comments

  • rip  On February 5, 2011 at 3:23 pm

    Hi,

    Perhaps you want to change “Sym(6)” in the title to “Sym(5)” ?

    • nbloomf  On February 5, 2011 at 3:43 pm

      Ahem… yes. 🙂

      Thanks!

  • kimochis  On October 5, 2013 at 2:28 pm

    why is order 4 possible?
    in case (–) (–) 1, the order is 2

  • kimochis  On October 5, 2013 at 2:29 pm

    why is order 4 possible?
    in case (a, b ) (c, d) 1, the order is 2

  • kimochis  On October 5, 2013 at 2:31 pm

    Sorry, I was wrong, order 4 is possible in the case (a,b,c,d) 1

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