Properties of the Dihedral Group D8
Not what you're looking for?
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).
Purchase this Solution
Solution Summary
We solve several problems involving the dihedral group D8, i.e. the symmetry group of a square.
Solution Preview
Please see the attachment.
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 ...
Purchase this Solution
Free BrainMass Quizzes
Probability Quiz
Some questions on probability
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.