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Permutation Groups

  1. Jul 8, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that if G is any group of permutations, then the set of all even permutations in G forms a subgroup of G.

    I am not sure where to start - I know there is a proposition that states this to be true, but I know that is not enough to prove this statement.
     
  2. jcsd
  3. Jul 8, 2011 #2

    tiny-tim

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    Hi PhysicsUnderg! :smile:

    Hint: if H is a subgroup of G, then for any a and b in H, the product ab must also be in H. :wink:
     
  4. Jul 8, 2011 #3
    Is it really that simple? lol This is what I was thinking, but I wasn't sure how to connect the idea to permutations. Can I just say "a is an even permutation" and "b is an even permutation" thus "a*b is also an even permutation"? Because, if this is true and if I assume that a and b are elements of H, then ab is an element of H and is an even permutation, so G has a subgroup of even permutations. Also, for H to be a subgroup, the identity element must be contained in H, as well as an inverse. How do you connect this to permutations?
     
  5. Jul 8, 2011 #4

    tiny-tim

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    Hi PhysicsUnderg! :smile:
    Yes!! :biggrin:

    Sometimes, maths really is that simple! :wink:
    You ask "Is the identity an even permutation? What is the inverse of an even permutation?" :smile:
     
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