# Thread: Is this a homomorphism?

1. ## 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)?

2. Originally Posted by 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)?

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

3. Originally Posted by tonio
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 $\displaystyle \langle a, b; \emptyset \rangle$.

4. Originally Posted by 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)?
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 $\displaystyle \phi(W)=1.$

5. So basically to check well-defined I need to make sure that the relations hold?