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Math Help - Need help getting started.. Abstract Algebra group problem

  1. #1
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    Need help getting started.. Abstract Algebra group problem

    Let (G,*) be a group. Define a new binary operation • on G by setting a • b = b * a for all a,b in G. Prove that (G,•) is a group
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  2. #2
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    Quote Originally Posted by tas10 View Post
    Let (G,*) be a group. Define a new binary operation • on G by setting a • b = b * a for all a,b in G. Prove that (G,•) is a group
    Well, how is being a group defined? - Once you remember that, you can just step through the group axioms and show that they hold for (G,•) as well, provided they hold for (G,*).

    For example: the associative property for (G,•) can be shown like this:

    (a•b)•c = c*(b*a) = (c*b)*a = a•(b•c)

    The first = holds, by definition of • in terms of *, the second = holds because (G,*) is a group, and the last = holds again by definition of • in terms of *.

    Now you need to show the existence of a neutral element and of inverse elements.
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    Would there be an identity element simply because (G,*) is a group?
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  4. #4
    Super Member Failure's Avatar
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    Quote Originally Posted by tas10 View Post
    Would there be an identity element simply because (G,*) is a group?
    Well, my guess is that the identity element of (G,•) is exactly the same as that of (G,*), so let's call the identity element of (G,*) simply e. Of course, that's just a hypothesis at this point, a lucky guess I hope. Now try to prove that this element e of G has the properties that are required if it is to qualify as an identity element of (G,•)!
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