The dihedral group D, is the simplest noncyclic group of order four D = {e,h,u,r) generated...

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The dihedral group D, is the simplest noncyclic group of order four D = {e,h,u,r) generated by rotations (r) and reflections (h, v). e is the identity element. (i) Construct the Cayley multiplication table for D. (ii) By proving that the map:9D(g) is a homomorphism, show that the following set of matrices is a representation of D. D(e)= = (1) D(h) = (1) D(v) = (0-1) D(T) = D(r)-( -(). 0 (iii) The above representation is reducible to one dimensional representations. Find out non-trivial and in-equivalent one dimensional representation for D. (iv) What would be the minimum dimension(s) of irreducible representations of the Groups of order 5? Give reasons to support your answer. The dihedral group D, is the simplest noncyclic group of order four D = {e,h,u,r) generated by rotations (r) and reflections (h, v). e is the identity element. (i) Construct the Cayley multiplication table for D. (ii) By proving that the map:9D(g) is a homomorphism, show that the following set of matrices is a representation of D. D(e)= = (1) D(h) = (1) D(v) = (0-1) D(T) = D(r)-( -(). 0 (iii) The above representation is reducible to one dimensional representations. Find out non-trivial and in-equivalent one dimensional representation for D. (iv) What would be the minimum dimension(s) of irreducible representations of the Groups of order 5? Give reasons to support your answer.