April 2011 WEATH3021 Group Theory Midterm test QUESTION 1 (a) Let G be a group and / be a subgroup of G. Define what is meant by a Normal subgroup of G and give one equivalent statement (3) (b) Let H and N be subgroups of G. Show that: () If G is an abelian group then AN is a subgroup of G. (2) (ii) If N normal in G then AN = NH and it is a subgroup of G. (6) (iii) If A and A are both normal in G and HAN = (e) then 125 he = mh for all he H, ne N. (4) 3 30 (iv) If H and N are both normal in G and HAN = (e) then H xN = HN where H X N - { (h, m ) | hCH. MEN ) (6) 3 36 (c) Let K be a subgroup of G and let / be a normal subgroup of G. Prove that NOK is a normal subgroup of A". (6) 3:42 . [27] QUESTION 2 ta ail rge (a) Let G be a group and let a be an element of the group. Define what is meant by: (i) The order of group G. # of ehweeds in C in cordiality of the group (ii) The order of the element a in G ."me why mistakes pose inleger it my ideably (iii) G the cyclic group generated by a. G ( a' In z f - ca> , jim ard (3) 45 (b) Prove that a subgroup of a cyclic group is cyclic. (6) 51 (c) State and prove Lagrange's Theorem for finite groups. (8) 59 (d) Let G be a group of order 6. Show that either G is cyclic or G = S, . (10) B.10 (e) List the subgroups of S, and the cyclic group C, and draw their lattice diagrams. (6) 4.16 [33] TOTAL - 60