I've just finished reading Chapter 6 of "Presentations of Groups," by D. L. Johnson.
The Details:
Quoting Johnson . . .
Definition 1: Let $\langle X\mid R\rangle$ be a finite group and put $F=\langle X\mid\rangle$, $\overline{R}=N$, so that $G$ is isomorphic to the factor group $F/N$; then to perform a coset enumeration on $G$ is simply to count the cosets of $N$ in $F$ (or $E$ in $G$), that is, to find $\lvert G\rvert$.
The group $E$ is understood to be trivial.
Quoting Johnson . . .
For each $$r=x_1\dots x_n\in R,$$ with $x_1\dots x_n$ a reduced word in $X\cup X^{-1}$, we draw a rectangular table having $n+1$ columns and a certain (for the moment unlimited) number of rows; thus
$$\begin{array}{c |c| c |c| c | c |c| c} & x_1 & & x_2 & \dots & x_n & \\ \hline 1 & & 2 & & \dots & & 1 \\ \hline 2 & & & & \dots & & 2 \\ \hline & & & & & & \end{array}$$
We begin by entering the symbol "$1$" in the first and last places of the first row of each table, the remaining places in the first rows being as yet empty. [ . . . ] Suppose the situation to be as in the above diagram with $2$ immediately to the right of $1$ and with $x_1\in X\cup X^{-1}$ lying between them. Now we put $2$ in the first and last places of the second row in each table and, wherever in any table $1$ lies to the left of an empty space with $x_1$ between the two spaces or to the right of any empty space with $x_1^{-1}$ between the spaces, we fill that empty space with a $2$.
So far so good (I think). I get the bit in italics, but where do the other $1$s come from?
Continuing to quote Johnson . . .
Similarly, if $2$ lies to the right (left) of an empty space with $x_1$ $(x_1^{-1})$ between the spaces, we fill in that space with a $1$. The idea behind this process (which we call "scanning") is that $1$ and $2$ correspond to $e$ and $x_1$ of $G$, so that we may write $1x_1=2$ and $2x_1^{-1}=1$.
Again, so far so good . . .
Having made sure that no more spaces can be filled in this way, we fill in an empty space with the new symbol $3$, begin a new row of the tables and scan as above, entering all possible $1$s, $2$s, and $3$s in accordance with the definitions of $2$ and $3$.
What definitions? Does he mean $2$ and $3$ corresponding with $x_1$ and $x_2$, respectively? I don't see how to do that if so.
Further . . .
Then fill a new empty space with the symbol $4$, begin the fourth row of the tables and scan again.
What is meant by "scan again"?
The Question:
Would someone explain to me the Todd-Coxeter coset enumeration algorithm, please?
Please help :)
NB: I've read the Wikipedia article on the Todd-Coxeter algorithm, but it doesn't illuminate things for me, unfortunately.