Indian Institute of Technology Mandi Department of Mathematics Abstract Algebra(Normal subgroups, external direct product, isomorphisms theorem for groups and fundamental theorem for finitely generated abelian groups) Tutorial, Date: 19/09/2023 Tutorial - 3 1. Find all the normal subgroups of the following groups : (a) Sn , permutation group on n symbols for n ̸= 6. (b) An , the alternating group. (c) Q8 , quartinion group. 2. Find at least two examples of groups that has subgroups isomorphic to Zn for all positive integers n. 3. Show that G ⊕ H is abelian iff G and H are abelian. State the general case. 4. If G ⊕ H is cyclic, prove that G and H are cyclic. State the general case. 5. Suppose that G1 ≈ G2 and H1 ≈ H2 . prove that G1 ⊕ H1 ≈ G2 ⊕ H2 . State the general case. 6. If a group has exactly 24 elements of order 6, how many cyclic subgroups of order 6 does it have ? 7. Let p be a prime. Find the number of subgroups of Zp ⊕ Zp of order p. 8. Show that Q, the group of rational numbers under addition, has no proper subgroup of finite index. 9. Suppose that G is a non abelian group of order p3 , where p is a prime, and Z(G) ̸= e. Prove that |Z(G)| = p. 10. If G is non-abelian, show that Aut(G) is not cyclic. 11. Show that there are two abelian groups of order 108 that have exactly one subgroup of order 3. 12. Let Sn denote the symmetric group on n symbols. The group S3 ⊕ (Z/2Z) is isomorphic to which of the following groups ? (a) Z/12Z (b) (Z/6Z) ⊕ Z2 (c) A4 , the alternating group of order 12. (d) D6 , the dihedral group of order 12. 13. Find the total number of non-isomorphic groups of order 122 . 14. Let G = GLn (R)and let H be the subgroup of all matrices with positive determinant. Prove or disprove that G/H ≈ {1, −1}. 15. Let G be a group. WOTF are true ? (a) The normalizer of a subgroup of G is a normal subgroup of G. (b) The centre of G is a normal subgroup of G. (c) If H is a normal subgroup of G and is of order 2, then H is contained in the centre of G. Best wishes Page 2