Discrete Mathematics and Group Theory

Discrete Mathematics and Group Theory
The following presentations represent groups of order less than or equal to 16:

G1 = <x, y | x8 = 1, y2 = x2 , xy = yx-1 >

G2 = <x, y | x2 = 1, y3 = 1, (xy)3 = 1>

G3 = <a, b, c, d, e | ab = c, bc = d, cd = e, de = a, ea = b>

G4 = <x, y | x8 = 1, y2 = 1, xy = yx5 >

G5 = <a, b | a4 = 1, b2 = 1 , ab-1 = ba-3>

Before beginning the questions below you should calculate the order of each of the groups G and produce the main table via implementation of the Todd Coxeter process. You will also need to record the names of each of the group elements expressed in terms of the generating symbols.

The descriptions below refer to exactly one of the groups described above (though the same group may correspond to more than one of the descriptions). Use the Todd Coxeter process and its subgroup variant, together with any theorems we have proved in the course notes to determine which of the groups corresponds to the following descriptions:

1. The group is cyclic.

2. The group is non-abelian and has no non-cyclic subgroups containing four elements.

3. The centre of the group is cyclic of order 4.

4. The group is abelian but non-cyclic.

5. The group has exactly three elements of order 2 and the centre of the group consists of the identity element only.

Your answer to this question should consist of relevant tables and a summary of the conclusions that can be drawn from them. You may also use any theorems from your notes or other sources provided they are accurately quoted.

The presentation and the summaries of your results will be taken into account in marking this assignment.

Question 2 (20 marks):

Use a hand implementation of the Todd Coxeter method to find

a) the order of the group

G = <a, b | a2 = b4 = 1, baba = 1>
b) the number of cosets of the subgroup generated by a-1b in the group
G above.

For part a) you should produce the main table and a list of relations found. You should indicate clearly in your main table the places at which definitions have been made and any bonus relations discovered. You should also include a list of the group elements.

For part b) you should indicate clearly any definitions and bonus relations that you use in your table. You must also write down a list of coset representatives (a transversal) for your subgroup.