I have read some material on Coset Enumeration. Unfortunately I could not follow the steps in Todd-Coexeter Algorithm, and also in Handbook of Computational Group Theory by Derek Holt. The problem is how we scan an element of subgroup and deduce the results by coset tables. How we make cost tables in GAP and also manualy. Could you please explain it with examples. Thanks.
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Are you trying to learn to do it by hand, or do you want to write a program? Johnson's book on finitely presented groups is probably the best I know of for the former. You mentioned you had some difficulty understanding Holt's Handbook but, really, I cannot think of anything better. There is also a book on computing with finitely presented groups by Charles Sims, which is similarly excellent. You could try studying the code for the coset enumerator built into GAP, which is open source. (James)
You may find to be useful Max Neunhoeffers' slides from the LMS short course on Computational Group Theory, see here - in particular, the Todd-Coxeter algorithm is in lecture 2, and there are also examples in GAP linked from slides. (Alexander Konovalov)