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**Swlabr** 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)?