Definition of generators in the context of groups as languages

Question

In the book "Word processing in groups" by Epstein et al. (p.28-29), the definition of generators begins with the following sentence:

Let $G$ be a group, $A$ an alphabet and $p \colon A \rightarrow G$ a map, which need not be injective.

My question is whether or not he assumes $A \subset G$. The confusion arises because on the one hand, he writes in the preceding paragraph:

Let $G$ be a group and $A\subset G$ a finite set of elements of $G$.

But later on, he writes in "Convention 2.1.2 (inverse generators)":

We call $\iota (x)$ the formal inverse of the generator $x\in A$, even though $x$ is an element of $A$, not of $G$.

which contradicts $A\subset G$.

Answer 1

No, $A$ is not assumed to be in general a subset of $G$. Formally, $\iota(x)$ could be defined as $p(x)^{-1}$. If $A$ is a subset of $G$, then $p$ is supposed to be the identity map.