Group Theory — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

1. (a) Prove that the order of a permutation of a finite set written in disjoint cycle form is the least common multiple of the lengths of the cycles. (7.5) (b) (i) Let S, denote the symmetric group of degree n. In Sz, find elements a and B such that a=2, |B|= 2 and laßl = 3. (ii) Let B e S, and B* = (2 1 4 • 35 6 7). Then find B. (3,4.5) (c) (i) Give two reasons to show that the set of odd permutations in S, is not a subgroup of (ii) Define even and odd permutations and show that the set of even permutations in S, is a subgroup of S. (3, 4.5) 2. (a) (i) Let a be an element in a group G such that la) = 15. Find all left cosets of (a) in (a). (ii) State and prove Lagrange's theorem. (b) Suppose that G is a group with more than one element and G has no proper, non-trivial subgroups. Prove that |G| is prime. (7.5) (c) Let C* be the group of non-zero complex numbers under multiplication and let H = {a + bi e C* la? + b2 = 1}. Give a geometrical description of the coset (3 + 4i)H. Give a geometrical description of the coset (c + di)H. (7.5) (a) (i) Let G be a group and H be its subgroup. Prove that if H has index 2 in G, then H is normal in G. (ii) If a group G has a unique subgroup H of some finite order, then show that H is normal in G. (3,4.5) (b) (i) Prove that a factor group of a cyclic group is cyclic. Is converse true? Justity your answer. (ii) Let G be a group and let Z(G) be the center of G. If G/Z(G) is cyclic, then show that G is Abelian. (3,4.5) (c) (i) Let ¢ be a group homomorphism from group G, to group G, and I be a subgroup of Gy. Show that if H is cyclic, then (H) is cyclic. (ii) How many homomorphisms are there from Zzo to Lg? How many are there onto lg? 4. (a) (i) Suppose that $ is a homomorphism from U(30) (ii) Let q be a homomorphism from a group G, to group G2. Show that d(a) = q(b) ift aker (b) (i) Is U(8) isomorphic to U(10)? Justify your answer. (ii) Show that any infinite cyclic group is isomorphic to the group of integers under addition. (3,4.5) (c) If o is an onto homomorphism from group G, to group G2, then prove that G,/Ker & is isomorphic to G,. Hence show that if G, is finite, then order of G, divides the order of G,. (7.5) (a) Let G be a group and let a e G. Define the inner automorphism of G induced by a. Show that the set of all inner automorphisms of a group G, denoted by Inn(G), forms a subgroup of Aut(G), the group of all automorphisms of G. Find (b) Prove that the order of an element in a direct product of a finite number of finite groups is the 1cm of the orders of the components of the element, i.e., 1g,, 82 ..., 8n) = 1cm (811, 1821.... Enl). Also, find the number of elements of order (c) Without doing any calculations in Aut (210s), determine how many elements of Aut(7,0s) have order 6. (7.5) 6. (a) For any group G, prove that G/Z(G) = Inn(G).