Find all possible orders of elements in Sym(7)

Find all numbers n such that S_7 contains an element of order n.


Note that the only possible cycle shapes of elements in S_7 are 1, (2), (2,2), (2,2,2), (3), (3,2), (3,2,2) (3,3), (4), (4,2), (4,3), (5), (5,2), (6), and (7). By a previous exercise, the order of a product of disjoint cycles is the least common multiple of the lengths of the factors. So the possible orders of elements in S_7 are (in increasing order) 1, 2, 3, 4, 5, 6, 7, 10, and 12. \blacksquare

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Comments

  • Victor  On September 17, 2011 at 7:12 pm

    why isnt 2*2*3 a possibilty

    2 cyc*2cyc*3cyle

    the total is 7 so we are still good 🙂

    • nbloomf  On September 17, 2011 at 9:06 pm

      Thanks!

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