Question

Qi Consider the group (D x Q, +), where Q Q = {(a,b)|a, b E Q}, and where addition is defined in the usual way by (a, b) +(c,d) = (a +c, b+d). So, for instance, (,-) € QxQ, and (2, - ) + (1, 1) = (1, 1). (a) What is the identity in this group? You do not need to justify your answer. (b) What is the inverse of the element (x, y) E Q? You do not need to justify your answer. (c) Is the group (D x Q, +) cyclic? Explain why / why not. (d) Is the group (x Q,+) isomorphic to the group (Q, +)? Explain why why not. (10 marks)

Answer 1

All answers are below thank you.

1 - he Group (exet where a xQ = 219,6) la, boh % The identity element in (GxG),+) is (0,0). lit. (a,b) + (0,0) = (a to, b+c) -(a,b). The incesse element in a xo į (-2,-4) where (29) 66XQ). Hot (24) + (-2,-4) = (2-2, 4-y) = (0,0). O caxa,t) iſ not cyclizi because a i nit cyali it has no generater. 10 (8x6 ) U (3-4 phfz + (@ 1) - yen fest. This ve desine map Q:(@x@,+) 76,+) by Diagb) = a.tb then conilder (a.be, Cad) taxa Q[ca,b)*(64)] = @[ca+c, b+d] = atc tbed a dca,b) + acced) dis homomorphins = af atb totd

- Now let (a,b), (c,d) taxa such that d(a,b) =accedy » atb = ctd * (a,b) = (eid) and d is one-one nap. I let a,bt ca then atbte. I ca,b) taxe such that acaib) = @atba a is onto also - dissatisly homemarphin, ohe, me and onto - din isomorphism. :. (axe,t) i isomorphiz to (@it) Thank you.