basing myself on suggestions I found in previous discussions, I have opted for the algebra book of Dummit.
However, I have now a little problem. On page 26 (3th edition) the authors say that
"presentations" give an easy way of describing many groups, but there are a number of subtleties that need to be considered. One of this is that in an arbitrary presentation it may be difficult (or even impossible) to tell when two elements of the group (expressed in terms of the given generators) are equal. As a result it may not be evident what the order of the presented group is, or even whether the group is finite or infinite.
Then they report two examples
$\qquad$$\qquad$$\qquad$$\qquad$ $<x_1,y_1 \;|\; x_{1}^2=y_{1}^2=(x_{1}y_{1})^2=1>$
saying that one can show this is a presentation of a group of order 4, whereas
$\qquad$$\qquad$$\qquad$$\qquad$ $<x_2,y_2 \;|\; x_{2}^3=y_{2}^3=(x_{2}y_{2})^3=1>$
is a presentation of an infinite group.
The problems I encountered are essentially two:
1) I have found vague the "definition" of "relations"(and consequently that of "presentation"), because it is set metamathematicalwise (at least this way it sounds to me), using the concept of "derivation/deduction". Infact, no method is exhibited to clarify and rigorously formalize the expression used in the "definition", so I find some trouble, the underlying ideas having been left up in the air (what is meant, for example, as they say "it may be difficult (or even impossible) to tell when two elements of the group are equal"? what's the corresponding mental mechanism? and what if the $S$ generators set has an exorbitant cardinality? Is perhaps (also) this a motive of difficulty to which the authors refer?);
2) with reference to the two examples I mentioned, I am not able to understand the link between these and what has been said just before: there may be some problems about assigning the order to the presented group (it has been said), but the two presentations in the examples are well defined in this sense (I say).
Best Answer
Let $\bar x$ be a vector of variables (e.g. $\bar x = a,b,c$).
Let $\bar R$ be a list of words formed from the variables in $\bar x$ (e.g. $aa$, $b^{-1}c^{-1}bc$).
The presentation $\langle \bar x \mid \bar R \rangle$ defines a group that is a quotient of [the free group on the symbols $\bar x$] by [the smallest relation on words of $\bar x$ that makes each word of $\bar R$ the identity].
A simple example is $\langle a \mid a^3 \rangle$, this defines the group $C_3$ because we start with the free group $\{\ldots, a^{-2}, a^{-1}, a, a^{2}, a^{3}, \ldots\}$ (this is isomorphic to $\mathbb Z$) and then quotient it out by the relation defined by $a^3 = 1$. That relation is the reflexive symmetric transitive closure of:
Throughout I used the term 'words', but it's really 'words plus inverses' but I don't have a name for that.