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Math Help - Homomorphism

  1. #1
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    Question Homomorphism

    Identify the kernel and whether or not the following mapping is 1-1 or onto
    1. G group, φ: G-> G defined by φ(a) = a^-1 for a is an element of G

    How can I find out if a homomorphism is 1-1 or onto
    Thanks
    Last edited by Godisgood; September 30th 2009 at 10:53 AM.
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  2. #2
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    Quote Originally Posted by Godisgood View Post
    Identify the kernel and whether or not the following mapping is 1-1 or onto
    1. G group, φ: G-> G defined by φ(a) = a^-1 for a is an element of G

    How can I find out if a homomorphism is 1-1 or onto
    Thaks
    ker \phi=\{x \in G:\phi(x)=e\}=\{e\}. The second equality obtained from the definition of \phi
    ker \phi=\{e\} iff \phi is one-to-one. However, you have to have the assumption that \phi is a homomorphism.
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  3. #3
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    1-1 and onto are two separate concepts, so there is no 'or' concept here.

    for 1-1: Show if φ(g1) = φ(g2) then g1=g2. This is very easy in this problem

    for onto: Show for ever g in G, there is an h in G such that φ(h) = g. This is again easy here as it follows directly from Group Axioms

    Quote Originally Posted by Godisgood View Post
    Identify the kernel and whether or not the following mapping is 1-1 or onto
    1. G group, φ: G-> G defined by φ(a) = a^-1 for a is an element of G

    How can I find out if a homomorphism is 1-1 or onto
    Thaks
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  4. #4
    Super Member Gamma's Avatar
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    I would also like to point out here that this mapping is not a homomorphism unless the group is abelian.

    \phi(ab)=(ab)^{-1}=b^{-1}a^{-1}
    \phi(a)\phi(b)=a^{-1}b^{-1}

    So this is a homomophism if anf only if a^{-1}b^{-1}=b^{-1}a^{-1}\Leftrightarrow aba^{-1}b^{-1}=1 \Leftrightarrow ab=ba for all a,b \in G. That is G is abelian.
    Last edited by Gamma; September 30th 2009 at 08:53 AM. Reason: tex error
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