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Homework Help: Identity element for group theory

  1. Jan 25, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that (S, *) is a group where S is the set of all real numbers except for -1. Define * on S by a*b=a+b+ab

    3. The attempt at a solution
    Well I know that i have to follow the axioms to prove this. So I started with G1 which is associativity. This one I got to work. Then G2 says i need to find an identity element. But this is where i got confused. I think that the identity element is 0. but i don't know if that is write. I don't really know how to find it. Then finally G3 is the inverse and I also got stuck here. I don't really know what the inverse would be. Can any one help.
  2. jcsd
  3. Jan 25, 2009 #2
    you just need to work from the definition

    so you want to find an element, call it "e", s.t

    a*e=e*a. (need to check this holds both ways)

    plug that into the definition for *. what i mean is replace the b with e and solve for e. try that. if it works one way you need to show that it works the other way.

    Just a note, if this group commutes or is abelian you only need to show one direction, but you would need to show commutativity.
  4. Jan 25, 2009 #3


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    Staff Emeritus
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    Gold Member

    There's an easy way to tell -- plug it in to see if it satisfies the identity.

    Yes you do; you're just being timid. You know the equations that the identity is supposed to satisfy... and you know how to turn it into an ordinary equation involving ordinary addition and ordinary multiplication of real numbers... and you know how to solve equations involving ordinary addition and ordinary multiplication of real numbers.... You know all the steps involved, and the path is direct, you just have to start moving.

    Again, same situation as the last question.
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