Question

1. (1) Describe all elements of order eight in Q/Z.

(2) Find all elements of infinite order in Q/Z.

(3) Identify Q/Z with a subgroup of C∗

Answer 1

: : EN ( Q,+)ů a commutative group. (2+) is a suligroup of (@+) since, (Q,+) is a commutative group, so every subgroup is normal . subgroup. :. ( 4+) n. a quotient group which is also commutatives: all elements of (0/+) in of the form & +2 where pe z det +2 € % 8.4.0 (62)-2. thin 3 (+2). – P+2 = 2 • P€2. ses. The order of this is?? au mā elementi of ours in (20+). in of in form, +2 , 4€2. 3 . . is an infinite group cent order of all elements is finite onder sinua, +2 (whenes, pe 2, ZEN and 9.9.4 (0,2)=1) 3. am : arbitrary element of (6/2+). 2 +2) =P*? = ? - order of 2+2 is a which is finite 10%+) has no elements of infinite onder (0) 4:Q,+) --- $(3611=1};}s (0%) defined by a $(2) = cos 242 tismaz j2016 Q. $() = cos say tisin zary . 4() = cos 20 (2+) + i sin 20 (***) .. =(65 262 +2 Sim xax)( Gas zaytisin 2ay) ... 40(2). ep (8) #x,yeQ. - 4 is a homomorphism. ... Kerep =*EQ I cpm) = 1} fre Q I (052*x +isin 3x =1} = {as2} - z.

I exists [ kerep =z is a normal subgroup of (Q,+)] .. et This homomorphism is onto homomorphism by est is omorphism Theorem, 1%,+) = ({3€¢18-1},"] Sinu (Rzec1 181=13.) in a subgroup of (65,-) :: (%,+) ů a subgroup of (C",)