MAT311-ASSIGNMENT 17 September 2018QUESTION 1LetGand Hbe groups and let G H =f(g; h ) :g2 G; h 2Hg. Dene amultiplication on G H by (a; b )(c; d ) = ( ac; bd )for all a; c2G and b; d2H.Further let G1=f(a; eH) :a2 Gg.(a)Prove that G H is a group under this multiplication. 5(b)Prove that if Gand Hare both Abelian then G H is Abelian.

3(c)Prove that G1is a subgroup ofG H. 4(d)Given that every group of order 6 is isomorphic to the cyclic group C6or the symmetric group S3, determineC2C3.3QUESTION 2 LetGbe a group and let Hand Kbe subgroups such that none of the sub-group is contained in the other.(a)Prove that HK is never a subgroup in G. 5(b)Prove that a group cannot be written as the union of two proper sub- groups. 5QUESTION 3 Prove thatH=1 a0 1 ja 2 IR:is an Abelian subgroup of GL(2; IR ). 10QUESTION 4 Consider the quotient groupQ=Z whose elements are of the form m n+Zwhere mand nare integers.

The representatives of Q=Z are rational num-bers in the interval 0;1) . Determine the order of(a) Q=Z . 2(b)each m n+Z. 3QUESTION 5(a)LetGbe an Abelian group and let Hbe a subset of Gdened as fol-lows, H=fa 2 G :o(a ) = n; n 2Zg. Prove that HG. 5(b)Prove that the additive group of rational numbers is not nitely gen- erated.

5QUESTION 6 (a)Determine whether the binary operationgives a group structure onthe given set. If no group results, give the axiom(s) that do not hold.(i) (2Z; ) given by ab= a+ b(ii) (C ; ) given by ab= jab j 3(b)Prove that a group Gis Abelian if and only if (ab )1= a1b 1for alla; b 2G.

5(c)Let Gbe a group and suppose abc=efor a; b; c 2G. Show that bca=e.2(d)Prove that if Gis a nite group of even order, then there is an elementa 6= 0 such that a2= e. 3(e)Let Gbe a group and suppose a2 G of odd order. Prove that aanda 1have the same order. Prove further that there is an element b2 Gsuch that b2= a. 5(f)In S3, give an example of two elementsaand bsuch that (ab )26= a2b 2.What does this example say about S3?3