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Math Help - showing that groups are isomorphic

  1. #1
    Super Member
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    showing that groups are isomorphic

    I have to show that the function gives the isomorphism of G with G', by showing that is a one to one function and is onto G'.

    From my notes,
    (1) if x = y, then e^x= e^y, so x=y. thus is one to one.
    (2) if r in R+, then
    (In r) = e ^ (In r) = r
    where ( In r ) in R. thus is onto R+.

    why does (1) and (2) prove that is one to one and is onto?

    thanks!
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  2. #2
    Senior Member Sampras's Avatar
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    To prove that a function is one to one, you show that  f(x) = f(y) \Rightarrow x = y or  x \neq y \Rightarrow f(x) \neq f(y) . Then to prove surjectivity, you want to show that every element in the codomain has a pre-image. For example, suppose  f(x) = x^2 and we define it from  \{1,2,3 \} to  \{1,4,9 \} . 1 has preimage 1, 4 has preimage 2, and 9 has preimage of 3. So it is surjective.
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