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Properties of the Dihedral Group D8

Let D_8 denote the group of symmetries of the square. Denote by a a rotation anticlockwise by ?/2 about the centre of the square, and by b a reflection through the midpoints of an opposite pair of edges.

(i) Verify that each rotation in D_8 can be expressed as a^i and each reflection can be expressed as a^(i)b, for i?{0,1,2,3}.

(ii) Verify the relations a^4=e, b^2=e and b^(-1)ab=a^(-1). Explain how these relations may be used to write any product of elements in D_8 in the form given in (i) above. Illustrate this with the example a^(3)ba^(2)b.

(iii) Find the conjugacy classes of D_8.

(iv) Show that the rotations in D_8 form a normal subgroup, H. Write down the distinct cosets Hg. Compute the multiplication table of the quotient group D_8/H. To which well-known group is G/H isomorphic? Is the subgroup generated by b normal in D_8?

(v) Viewing the square in the real plane, centred at the origin, write down the 2Ã-2 matrix ?(a) which represents the rotation a and the 2Ã-2 matrix ?(b) which represents the reflection b. Check that
?(a)^4=I_2
?(b)^2=I_2
?(b)^(-1) ?(a)?(b)=?(a)^(-1).
(This shows that you can define a homomorphism ?:D_8?GL(2,R) by letting ?(a^i b^j )=?(a)^i ?(b)^j.)

(vi) By labelling the corners of the square or otherwise, write down the homomorphism ?:D_8?S_4, verifying that
?(a)^4=id
?(b)^2=id
?(b)^(-1) ?(a)?(b)=?(a)^(-1).

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Let D_8 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.
Verify that each rotation in D_8 can be expressed as a^i and each reflection can be expressed as〖 a〗^i b, for i∈{0,1,2,3}.
Each element of can be represented as a permutation of the vertices of the square, which we can label as 1, 2, 3, and 4 going anticlockwise. In this representation we have and , where the vertices 2 and 4 are fixed by the reflection corresponding to b. From this permutation representation we see that is a rotation about the centre by , is a clockwise rotation by , and is the identity (no rotation). We also have , which is a reflection about the axis passing through the centre of the square and parallel to the edges 14 and 23, is a reflection about the axis passing through vertices 2 and 4, and is a ...

Solution Summary

We solve several problems involving the dihedral group D8, i.e. the symmetry group of a square.

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