I start my PhD in Mathematics this October and I'll be working closely with presentations, so a detailed answer aimed at that level would be ideal.
I want to understand the beginning of Section 2.3 of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by W. Magnus et al. In particular, I want to understand what a rewriting process is.
The Definition:
[L]et $G$ be presented [by $$\langle a_1, \dots, a_n\mid R_\mu(a_\nu),\dots\rangle]$$ and let $H$ be the subgroup of $G$ generated by the words $J_i(a_\nu), \dots$. Then a rewriting process for $H$ (with respect to the generators $J_i(a_\nu)$) is a mapping $$\tau: U(a_\nu)\to V(s_i)$$ of words $U(a_\nu)$ which define elements of $H$, into words in the symbols $s_i$, such that the words $U(a_\nu)$ and $V(J_i(a_\nu))$ define the same element of $H$. (The symbol $s_i$ will be the generating symbol used for $J_i(a_\nu)$ in the presentation of $H$ which we shall obtain.)
I don't understand this; in particular, I don't understand the example given just after it.
The Example.
For example, let $G$ be the free group on $a$ and $b$, and let $H$ be the normal subgroup generated by $b$. Then $H$ is generated by $b$ and its conjugates by powers of $a$ [ . . . ]. Let $$J_k(a, b)=a^kba^{-k}.$$ A word $W(a, b)$ defines an element of $H$ if and only if the exponent sum of $W$ on $a$ is zero. Indeed, if $$U(a, b)=a^{\alpha_1}b^{\beta_1}a^{\alpha_2}b^{\beta_2}\dots a^{\alpha_r}b^{\beta_r}\tag{1}$$ has zero exponent sum on $a$, then $U(a, b)$ and $$(a^{\alpha_1}ba^{-\alpha_1})^{\beta_1}\cdot(a^{\alpha_1+\alpha_2}ba^{-\alpha_1-\alpha_2})^{\beta_2}\cdot\dots\cdot (a^{\alpha_1+\alpha_2+\dots +\alpha_r}ba^{-\alpha_1-\alpha_2-\dots -\alpha_r})^{\beta_r}$$ define the same element of $H$. Hence the mapping $\tau$ which sends the word $(1)$ into $$s_{\alpha_1}^{\beta_1}\cdot s_{\alpha_1+\alpha_2}^{\beta_2}\cdot\dots\cdot s_{\alpha_1+\alpha_2+\dots \alpha_r}^{\beta_r}$$ is a rewriting process for $H$.
The Problem.
Would someone explain this to me, please?
Thoughts.
There's nothing much to add really. I suppose I get stuck when the $s_i$s are introduced both in the definition and the example. The example makes sense in that I get the conjugation bit and the zero exponent sum; I even get the substitution of the $s_i$s. What I'm struggling with is how it fits the definition and what the $s_i$s are doing.
The first sentence I don't understand starts
Then a rewriting process for $H\dots$
I hope I've provided enough context for this to be on topic.
Please help.