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Thread: Is this a homomorphism?

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

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

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

    such that $\displaystyle G = \langle A, B \rangle$ and $\displaystyle W_i(A, B) = 1$ in $\displaystyle G$ for all $\displaystyle i$, then is $\displaystyle G$ 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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    Quote Originally Posted by Swlabr View Post
    I have a group $\displaystyle G = \langle a, b; W_1(a, b), W_2(a, b), \ldots \rangle$ ($\displaystyle W_i(a, b)$ are words on a, b and are the relations) and define a function,

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

    such that $\displaystyle G = \langle A, B \rangle$ and $\displaystyle W_i(A, B) = 1$ in $\displaystyle G$ for all $\displaystyle i$, then is $\displaystyle G$ 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, $\displaystyle a\mapsto ab$ $\displaystyle b\mapsto b$ in the free group $\displaystyle \langle a, b; \emptyset \rangle$.
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    Quote Originally Posted by Swlabr View Post
    I have a group $\displaystyle G = \langle a, b; W_1(a, b), W_2(a, b), \ldots \rangle$ ($\displaystyle W_i(a, b)$ are words on a, b and are the relations) and define a function,

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

    such that $\displaystyle G = \langle A, B \rangle$ and $\displaystyle W_i(A, B) = 1$ in $\displaystyle G$ for all $\displaystyle i$, then is $\displaystyle G$ 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 $\displaystyle W = 1,$ where $\displaystyle W$ is some word in $\displaystyle G,$ then $\displaystyle \phi(W)=1.$ but if $\displaystyle W=1,$ then $\displaystyle W $ is in the subgroup

    generated by $\displaystyle W_i(a,b)$ and thus $\displaystyle \phi(W)$ would be in the subgroup generated by $\displaystyle W_i(A,B)$ and so $\displaystyle \phi(W)=1.$
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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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