I've just finished reading Chapter 6 of "Presentations of Groups," by D. L. Johnson.
I'm stuck trying to understand the tables in Example 2 of that chapter. (I have linked to the tables.)
In the first line of the second table, we have $1a4b5a2b^{-1}1$. Why is $5$ there and not in the second line of the first table, after $2$?
I follow the construction of the tables besides that bit. Is it arbitrary; that is, could I get an equivalent result if I had put the symbol $5$ after the $2$ in the second line of the first table?
Please help :)
Yes you can make definitions in any order that you please, except that you must ensure that every definition $kx_i$ (and also $kx_i^{-1}$) is eventually made. The biggest problem in coset enumeration is to decide which definition to make next, and the choices you make can result in a big difference to how long the process takes.
There are two main strategies that are used in computer implementations, and in the best implementations you can mix these two strategies in any specified proportion.
The first is to make all definitions $kx_i$ and $kx_i^{-1}$ in order of increasing $k$, and increasing $i$ for fixed $k$.
The second is to make definitions that fill in the rows of the relator tables in order. I think that is the policy being used in the example you mentioned. The definition $4b=5$ is made to complete the first row ot the two relator tables.
There is more scope for use of experience and judgement when doing it by hand. You may choose to make definitions that you can see are going to result in lots of deductions, but when you do that you must ensure that all possible definitions are eventually made.