Determining conjugacy classes

Question

Problems:

Let D₈ denote the group of symmetries of the square. Denote by a a rotation anticlockwise by π⁄2 about the centre of the square, and by b a reflection through the midpoints of an opposite pair of edges.

1. Verify that each rotation in D8 can be expressed as aⁱ and each reflection can be expressed as aⁱ b, for i∈{0,1,2,3}.

2. Verify the relations a⁴=e,b²=e and b^-1 ab=a^-1. Explain how these relations may be used to write any product of elements in D₈ in the form given in (i) above. Illustrate this with the example a3 ba2 b.

3. Find the conjugacy classes of D₈.

4. Show that the rotations in D₈ form a normal subgroup, H. Write down the distinct cosets Hg. Compute the multiplication table of the quotient group D₈⁄H. To which well-known group is G⁄H isomorphic? Is the subgroup generated by b normal in D₈?

5. Viewing the square in the real plane, centred at the origin, write down the 2×2 matrix ρ(b) which represents the rotation a and the 2×2 matrix ρ(b) which represents the reflection b. Check that

                                                     p(a)⁴ = 1₂
                                                     p(b)² = 1₂
                                                     p(b)^-1p(a)p(b)^-1

This shows that you can define a homomorphism p:D8→GL(2,R) by letting p(aⁱb^j) = p(a)ⁱp(b)^j)

6.By labelling the corners of the square or otherwise, write down the homomorphism

φ:D₈→S₄, Verifying that

                                             φ (a)_⁴ = id

                                             φ (b)_² = id

                                             φ (b)_^-1 φ (a) φ (b) = φ (a)_^-1