Group Theory and Applications — DU B.Sc. (Hons) Physics Previous Year Question Papers

2. Attempt FIVE questions in all. 1. Answer any SIX out of the following seven questions: (a) Show that the subset G = {1, -1, i, -i} of the set of complex numbers is a group under the usual complex multiplication. (b) If a group contains elements a and b such that order of element a; O(a) = 4, and order of element b; O(b) = 2 and cz3Zt = ba, find O(ab). (c.) Let G ± Z8 is a set of integers modulo 8 and H = {0, 4}. Find all left cosets of H in G. (d) Prove that if a group G has order i4, then it must be abelian. (e) State Lagrange's Theorem for a group. (i) Check whether the set G = {1, 2, 3, 4, 5} is a group with respect to multiplication modulo 6 or not. (g) Twhat is a Lie algebra? State the Jacobi identity. (a) Let Z denote the set of all integers. Prove that (Z, *) forms an abelian group under the binary operation defined by cz*b±cz+Z)+1, forallcz,bEZ. (b) Show that H = {0, 2, 4} is a subgroup of the group G = {0,1, 2, 3, 4, 5} under addition modulo 6. (c) Consider the dihedral group D4 ± {e, 7~, 72, 7;, s', s'r, s'72, s73}, representing the symmetries of a square. Determine whether D4 is a cyclic group. (a) Let G = S3, the symmetric group on three elements: Construct tlie cayley table of s3. Whether s3 is an abelian group or not. (b) What is a group isomoxphism? Show that the groups ({1, -1}, X) and ({0,1}, are isomorphic. (a) Detemine whether the following permutations are even or odd: (b) Define a representation of a finite group. Consider a representation of a group G acting on a 3-dimensional space with basis {ei, e2, e3} . Under a group element R, the basis vectors transform as: ReL= e\, Re2=e2, Res--Te3 (i) Write the matrix representation of R (ii) Find its character x(R). (iii ) Detem'ine whether the representation is reducible or irreducible. (a) Consider the set SO(2)± (:;:8 -c::nGG) Show that SO(2) is a Lie group and find its dimension. (b) What is SU (2)? Explain its connection with quantum spin. For two spin-2± particles, detemine the pos.sible total spin states and classify them according to their total spin. (a) Distinguish between point groups, Bravais lattices, and space groups in crystallography. Explain how each of these is used to describe the symmetry of crystals. (i) Detemine all conjugacy classes ofG. (ii ) Find the number and dimensions of all irreducible representations of G. (iii) Construct the complete character table of G. (iv) Verify the orthogonality relations of the characters.