Question

General exercises on Cossets and Lagrange's Thootem How Find the cosets of H in the group G when H=487 and G=(7+). Hiw 1 (ii) H = 4327 and G= (Zug ) (ii) H={(7,3), (737),(7,3)} and G = ( ZXZ,,D) GV) H=<x> and G=<x,y:**-1,x!:y?, j'xy-z'> Find [G:H] when: i) H= {(1,1)> and G=(ZxZy, lid H= (a) and G=<a, b: 22=6²=(ab) = e> Chi H=<4> and G = 1 Zız@) (iv) H={(123) > and G = S, (3). I£ G = {Z, *) , where * is defined as follows a*b= a +(-19 6 Haabega If H = <17, find lis. The elements of H il H#2 and 2## (4) If G is a group, 1G1 = 9, prove that G is cyclic of x'se #zeg. (5) If G is afinite group and H, K<G, prove that |HK| = |AT|KI (6) Find all the subgoups of G when: (i) G = D. = <a,b; Q4 +2=63, 6-'ab=a's (i) G = fly THOK!

1,2,3,4,5 and 6

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Answer 1

2 2 = { **&, 1* 27 = , -1] 2xH = £g*0, 2*13 = {2,39 16 = 9=300 Now, G i abelian. since or = p2 . of or is cyclic we are done It not there does not any element such that 0 (H) = 9 OCH Ol of = 3 or I & HEG. Hence and oce=1 only e is identy and hence 3 = e tutur. 6 (a) G = D8 - Lab ib) De , - <a, b : a = e = 6² brabal} sub groups of Dg are. <a>, <a>, <b>, <ab>,<a6²>, 69²6² W = Au ola) = 12 elkelin group) <(1,2,3)>, < (1,2,4) > Lall, 3, 4) >,<(2, 3, 4>. Ay. Ay does not have subgroup of erder G. O (HK) = OCH OCK) OCHRK) HK = { hk : hEH, KEK? et nk = n'k for some hhEH & K,KEK. then in th=k'ktu since HSG WHEH and KLG K'KIEK NE HAK. ot follows that number of repetitions of given element hk CHK is ocHnKI Thus OCHK) o (HNK) = OCH ock) ar HKI - HIKI \ НАК

i H= <83 and un= (2,+). H+1, H 2, H+3 H+4, 145, H+6, HAT, H H= <387 and = (248, ) casets of Hin u are l H, HAI, Hož H03, HB4 Ho5, HAG, HOT, HAS Hoa, Holo, Holi, HoR, 1613, Hol4, H015, HANG, H = {(oo), lo, i, o, 2, 3 and or= {Zgx Z₂, 6) HAIT, o), H (T, TJ, HATI H = <sphere and or=<H, Y; "=1 #²ye, yosey=des Ha, sh, Hy, YH, Y²H = x²H = H " H= C(1,1> and v= (Za X Zy , ) (H) = 8 ,014)=8 Now & GitH] = C = 8 = 1 (1) H=La> and wala, bi a²6²= (ab = ez [ : HJ = (a 3 = 3 H=<4> and or= ( 212 ) CH) = 3 014) = 12 • [Git] =4 = < (1,2,3) and G=83 OCH) = 3 and olul) = 6 [Gil] = 2 = (2 ) axb=a+(-) va, btw and H= <i> elements of H- Hd H= { 1, IKI, 1*1*1, 1X*1x1 = 10,1 = { 1,03