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Homework Help: Abstract Algebra - Subgroup of Permutations

  1. Sep 28, 2011 #1
    1. The problem statement, all variables and given/known data
    A is a subset of R and G is a set of permutations of A. Show that G is a subgroup of S_A (the group of all permutations of A). Write the table of G.


    Onto the actual problem:
    A is the set of all nonzero real numbers.
    [itex]G={e,f,g,h}[/itex]
    where e is the identity element, f(x) = 1/x, g(x) = -x, h(x) = -1/x

    Would this be the right way to do it?

    For each combination of elements in G (call the elements a,b) I need to show
    a*b is in G

    I also need to show that the inverse of a is in G.

    Here is where I get confused, I'll start with with a = e:

    ee = e
    ef = f
    eg = g
    eh = h

    Okay, that is all good, now letting a = f:
    fe = e
    ff = e
    fg = h
    fh = g

    now a = g:
    ge = g !
    gf = h
    gg = g !
    gh = e

    now a = h:
    he = h
    hf = g !
    hg = f
    hh = g !

    What am I doing wrong here?
     
  2. jcsd
  3. Sep 28, 2011 #2
    The following are wrong:

    fe=e
    gg=g
    gh=e
    hg=f
    hh=g

    What did you do to obtain those?
     
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