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

Ufiique paper code ; `235,2013601 Durat'ion : 3 Hours Maximulm Marks : 90 All parts of a question are to be a.ttempted together. Use ofcal.cula,tor is not, allowed. .1.. (a) L.et a group G act Qn ,a set A. Show that the stabilizer of an element a € (b) Let a group G act on itself by left multiplication. Find the kernel of the action. (c) Prove that if A and a are subsets of G with A E a, then CG(a) is a subgroup of CG(A) where CG(A) denotes the centralizer of a subset A in G. (d) Show that the symmetric group S3 is solvable. (e) Let P = (1 2 3 4 5) e S5. Find an element i € S5 such that Tpt-I = p2 ® Prove that the product of two characteristic subgroups is a characteristic stibgroup. (g) Define a nilpotent group. Prove that every abelian group is n.ilpotent. 2. (a) Prove that actions of a group G on a set A and the homomolph]isms from G into the symn|etric group SA are in bijective correspondence. (b) (c) Prove Lagran8e's Theorem for finite groups using Group Actions. 1, 2, 3, 4 respectively. Under this labelling. and the action of,left multiplication by the group element, give the permutation representation associated to this action. 3. (a) (b) (C) Let G be a group, let H be a subgroup ofG and let G actbyleftmultiplication on the set A of left cosets of H in G. Let IIH be the associated permutation representation afforded by this action. Then prove that (i) G acts transitively on A ,< (ii) the s.tabilizer in G of the point lH e A is the subgroup H (iii) the ke`rnel of this action is rlx€G xHx-I and Ker(HH) is the largest normal subgroup of G contained iri H. Find all conjugacy classes of D8, the dihedral-group of order 8. Show that |Csn(12)(34)I = 8(n -4)! V n < 4. Determine the elements in this centralizer exp licitly. 4. (a) a) (c) Use sylow's theorem to determine ifa groupo.forder 105 is simple or not. Show that a group of order 56 is not simple. Prove that if G is a group of order 231 then Z(G), the center of G, contains a Sylow 11-subgroup of G and a Sylow 7-subgroup tis normal in G. 5. (a) (b) Let G be a group of order pq where p and q are primes, p < q and p does not divide (q -1). Then prove that G is a cyclic group. Hence dedu.ce that a group of order 33 is cyclic. Prove that there is no simple group of order 280. I (c) State and prove Index Theorem. Prove that a group of order 72 is not simple. 6. (a) (b) (c) Let p be a prime. Prove that ifG is a non-abelian group.oforderp3, then the commutator subgroup G' is equal to the center Z(G). Let p and q be primes. Show that any group of order p2q is solvable. Let G be a finite nilpotent group. Prove that no proper subgroup H of G is equal to its normalizer N(H).