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Composition of functions and stuff

  1. Sep 4, 2009 #1
    1. Show that the set {f:R-{0,1}[tex]\rightarrow[/tex] R-{0,1}}, of functions under composition, is isomorphic to [tex]S _{3}[/tex]
    [tex]f_{1} = x[/tex]
    [tex]f_{2} = 1 - x[/tex]
    [tex]f_{3} = \frac {1}{x}[/tex]
    [tex]f_{4} = 1 - \frac {1}{x}[/tex]
    [tex]f_{5} = \frac {1}{1 - x}[/tex]
    [tex]f_{6} = \frac {x}{x - 1}[/tex]


    2. Relevant equations



    3. The attempt at a solution

    I don't really understand what the problem is asking
     
  2. jcsd
  3. Sep 5, 2009 #2

    lanedance

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    Homework Helper

    hey polarbears - there might be a smarter way, but I would start by having a look at the group S3, eg. all permutations of a set of 3 elements & see if you can find a 1-1 correspeondance between elements of S3 & the functions you are given, that is preserved under multiplication (in this case composition of functions)

    for example, it should be clear that:
    f1(fn(x))= fn(x), for any n, which makes it a good candidate for the identity element

    info on S3 is here, have a look at the mult table in particular
    http://groupprops.subwiki.org/wiki/Symmetric_group:S3
     
    Last edited: Sep 5, 2009
  4. Sep 5, 2009 #3
    Is this a question from the Gilbert book?
     
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