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Math Help - Is this a homomorphism?

  1. #1
    MHF Contributor Swlabr's Avatar
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    Is this a homomorphism?

    I have a group G = \langle a, b; W_1(a, b), W_2(a, b), \ldots \rangle ( W_i(a, b) are words on a, b and are the relations) and define a function,

    \phi: G \rightarrow G
     a \rightarrow A
    b \rightarrow B

    such that G = \langle A, B \rangle and W_i(A, B) = 1 in G for all i, then is G a surjective endomorphism (homomorphism)?

    Basically, if I stipulate that the generators are sent to other generators and the relators still hold, then is it a homomorphism (the surjection easily follows)?
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  2. #2
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    Quote Originally Posted by Swlabr View Post
    I have a group G = \langle a, b; W_1(a, b), W_2(a, b), \ldots \rangle ( W_i(a, b) are words on a, b and are the relations) and define a function,

    \phi: G \rightarrow G
     a \rightarrow A
    b \rightarrow B

    such that G = \langle A, B \rangle and W_i(A, B) = 1 in G for all i, then is G a surjective endomorphism (homomorphism)?

    Basically, if I stipulate that the generators are sent to other generators and the relators still hold, then is it a homomorphism (the surjection easily follows)?

    This is weird (for me): G is generated by a,b...and also by A, and B? What did you mean exactly: a suppsoed homomorphism between two different groups G, G' or an endomorphism of G?

    Tonio
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by tonio View Post
    This is weird (for me): G is generated by a,b...and also by A, and B? What did you mean exactly: a suppsoed homomorphism between two different groups G, G' or an endomorphism of G?

    Tonio
    Endomorphism. So, for example, a\mapsto ab b\mapsto b in the free group \langle a, b; \emptyset \rangle.
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  4. #4
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    Quote Originally Posted by Swlabr View Post
    I have a group G = \langle a, b; W_1(a, b), W_2(a, b), \ldots \rangle ( W_i(a, b) are words on a, b and are the relations) and define a function,

    \phi: G \rightarrow G
     a \rightarrow A
    b \rightarrow B

    such that G = \langle A, B \rangle and W_i(A, B) = 1 in G for all i, then is G a surjective endomorphism (homomorphism)?

    Basically, if I stipulate that the generators are sent to other generators and the relators still hold, then is it a homomorphism (the surjection easily follows)?
    sure it is. the only thing you need to prove is that your map is well-defined, i.e. if W = 1, where W is some word in G, then \phi(W)=1. but if W=1, then W is in the subgroup

    generated by W_i(a,b) and thus  \phi(W) would be in the subgroup generated by W_i(A,B) and so \phi(W)=1.
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  5. #5
    MHF Contributor Swlabr's Avatar
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    So basically to check well-defined I need to make sure that the relations hold?
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