Consider a group $G$ generated by its subset $S$, i.e. $G=\langle S\rangle$. Then let $\phi:F(S)\rightarrow G$ be the corresponding surjective homomorphism from the free group $F(S)$ and $H=\{\phi^{-1}(g)|g\in G\}$ is the collection of preimages. Suppose the transversal is given, i.e. some set $X$ containing exactly one element from each set in $H$. Equivalently, $X$ consists of elements from $G$ written as some words in $F(S)$.
Now, let $R$ be such a set of the relations that we can logically deduce any word $w\in F(S)$ to one of the words from $X$. It's not hard to show that this condition is equivalent to the possibility to say whether $w$ is equal to the identity $e$ (if it's the case). Thus, $R$ allows us to define $ker(\phi)$. That's why $G$ must be defined uniqely by $S$ and $R$. My questions are the following:
- Can we say that $\langle S,R\rangle$ is the presentation of $G$, i.e. the normal closure $N(R)$ is equal to $ker(\phi)$?
- Given some presentation $\langle S,R\rangle$, can we say $R$ has the property described above, i.e. deducing in some finite number of stepts we will conclude that some word $w\in F(S)$ is equal to $e$ (if it's really so)?
Some clarifications. I've used here the phrase "logically deduced" assuming a bit of model theory. Actually, if we consider $R$ as an additional set of axioms why not to ask what these axioms help us to prove by the logical means.
Thanks in advance for your answer.
P.S. I'll be also grateful if you attach some links or name of a textbook where I can find the corresponding proofs.