4. Following are incomplete tables of 'Chains' and 'Links' for the Todd-Coxeter algorithm, applied to the presentation: (A, B | A4 = AB2 = 1, ABA= B). (We write b for B-1.) B b 1A2A3A 4A1 1A 2A 3 B B1 1A2B A551 1 2 5 2A3A 4A1A2 2A3A4B B2 2A 3B A 62 2 3 3A 4A1A2A3 3A 4A1 B5 B3 3A 4B A 53 3 4 4A1A2A3A4 4A1A2B B4 4A1B5A 64 4 5A6A A A5 5A6A B1B5 5A 6B 2A 365 5 6 3 1 6A A A5A6 6A A B B6 6A BA 56 6 5 a) Fill in any links that may not yet have been written into the 'Links' or 'Chains Tables, and any subsequent new links that may result. b) Choose an appropriate link for the 7th element; process any chains that this link affects, and any further links that result. c) Continue the Todd-Coxeter algorithm with this presentation, until the Links' table is full, and all 'Chains' are processed. How many elements are there in this group? d) Identify each of the elements of the group as a short product constructed from the generators A and B. (Choose 2 or more products for some elements, in such a way that it is clear that the group is complete, and that the original relations hold.) What is the order of the generator B? Give the standard name for this group, if you know it. 4. Following are incomplete tables of 'Chains' and 'Links' for the Todd-Coxeter algorithm, applied to the presentation: (A, B | A4 = AB2 = 1, ABA= B). (We write b for B-1.) B b 1A2A3A 4A1 1A 2A 3 B B1 1A2B A551 1 2 5 2A3A 4A1A2 2A3A4B B2 2A 3B A 62 2 3 3A 4A1A2A3 3A 4A1 B5 B3 3A 4B A 53 3 4 4A1A2A3A4 4A1A2B B4 4A1B5A 64 4 5A6A A A5 5A6A B1B5 5A 6B 2A 365 5 6 3 1 6A A A5A6 6A A B B6 6A BA 56 6 5 a) Fill in any links that may not yet have been written into the 'Links' or 'Chains Tables, and any subsequent new links that may result. b) Choose an appropriate link for the 7th element; process any chains that this link affects, and any further links that result. c) Continue the Todd-Coxeter algorithm with this presentation, until the Links' table is full, and all 'Chains' are processed. How many elements are there in this group? d) Identify each of the elements of the group as a short product constructed from the generators A and B. (Choose 2 or more products for some elements, in such a way that it is clear that the group is complete, and that the original relations hold.) What is the order of the generator B? Give the standard name for this group, if you know it
