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Math Help - Commutative Isomorphic?

  1. #1
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    Commutative Isomorphic?

    Suppose that (S, \bullet) and (T,*) are isomorphic. Show that (S, \bullet) is commutative if and only if (T,*) is commutative.

    Suppose that (S, \bullet) is associative. Is (T,*) associative? Prove your statement.

    I am having problems with the above question.
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  2. #2
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    Let f be an isomorphism between S and T

    let s,t\in T . Given that S is commutative, we want to show that st=ts.
    Since f is isomorphism, f is onto. So there are a,b such that f(a)=s and f(b)=t

    Now

    st=f(a)f(b)
    =f(a\bullet b) by definition of homomorphism
    =f(b\bullet a) since S is commutative
    =f(b)f(a) again by homomorphism
    =ts

    For associativity you may try by yourself first
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  3. #3
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    Ok this is what i got can u tell me if im right?

    If we suppose that (S, \bullet) is assiociative and let x,y,z \inT. Since f is isomorphic f is onto, so there are a,b,c such that f(a)=x, f(b)=y, f(c)=z

    x*(y*z)=f(a \bullet (b \bullet c)) since S is a homomorphsim.
    = f((a \bullet b) \bullet c) since f is commutative.
    =f(a \bullet b) * f(c) since it is a homomorphism
    =(f(a)*f(b))*f(c)
    =(x*y)*z
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  4. #4
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    Quote Originally Posted by I Congruent View Post
    Ok this is what i got can u tell me if im right?

    If we suppose that (S, \bullet) is assiociative and let x,y,z \inT. Since f is isomorphic f is onto, so there are a,b,c such that f(a)=x, f(b)=y, f(c)=z

    x*(y*z)=f(a \bullet (b \bullet c)) since S is a homomorphsim.
    = f((a \bullet b) \bullet c) since f is commutative.
    =f(a \bullet b) * f(c) since it is a homomorphism
    =(f(a)*f(b))*f(c)
    =(x*y)*z
    Becareful, don't skip steps
    Let me do first
    x*(y*z)=f(a)*(f(b)*f(c) ) just relacing x with f(a) and so on
      =f(a)*(f(b\bullet c)) definition of homomorphism

    Now try to continue
    Edit: well you don't need to because that two steps above is exactly the steps that are missing
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  5. #5
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    Thank you very much i think i finally understand this.
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