Question

please show your steps and your work clearly thank you!

2. Find all the subgroups of D4. Which subgroups are normal? What are all the factor groups of D4 up to isomorphism? Find all the subgroups of the quaternion group, Q8. Which subgroups are 3.

Answer 1

Sulogoroup oup of difedexel gustoup OCOW= 8 = 2x4 (2n)'. Dy=<q, b: 04= 6* = (46)=es Kea, a brab, alb, q3b> Reflection. 2. a a These are Rotation Now Bulgacoupa Seat Hi=&et H2 = q e, a, 9², 937 <a:a4=e7 Ha = q e, az <Q : (02)2 = ex. (02)4=g2. H4 = {e, be, <b: 6²=e> b= 6-1 H5 Xe, aby <b: (ab) ² = e> (ab) = (ab) H6 = {e, a²beca, b: (9²b) ² = ey 92b = 626) Hy = e, q3b4, <a, b, Q3b) = 2, 936=636)-7 Horte, 22, b, 4364TQ, b bea, abea I All are self inu] Hasle, 22 nb, 436% Kas, ab: (a)=(abso HiO D4 Dy his all possible to subgroups. since, total number of subgroups of Dy must be equel to sub quoup of D4 E no. of the divisor ofn + Oum of those divisor. no 3 + = 10 and we found all thope 10 šuleqroups. 1, 2, 4, ou div of 4 (1+2+4, sum of them

Now Next is Normal subgroups of Du Remember Dy is non-abelian. Note ③ aH = {ah Yah 1 h EdY left conet Ha = Y hal hery •Right copet - where his subgroup of a. in avelian que aH=19 group iwe when in non-abelian, aH= Ha neill say Hils normel subgroup of group a. H49. ② A sulegroup of index 2 is always normal Now evaluting each subgroup and finding which is nomal. 4 Hi neill be nomel, identity will commute, * H₂ = Y e, 9, 92, a 37, index of H2 = [4 : H₂] Qoca = 8 -2 O(H2) This is also nomal. e H₃ = {e, a2y, this is centre and centre is Nommel Guelquoup

949-1 sed .EH 501 * Hy = Ye, by <by since, Hy is generated by b. leta, quickly find is it normelor not using property Hii normed in a, if for oog felt gf G. We have where gH g t = lghg 1: het? ut h = b.CH and g= ab EG fab) "ab] then cab)(b)(ab) = abab = a²b & 4 [ 6² = 0 Hence * Hs Ye, aby again, using above mpeter h = ab EH and 2²6 € G. g- we have (936) Cab? (a3b). 4b2a3b = Zabt | C93b). 36 3 ea b = a3 b & P Not nomel OR HE = Ye, 92b} = Lazby et h= 926 EH g= a b e a (93b)(?) (236) = 0.93% H Not nomel. at Hy = {e, 486) 02b) =b

let h = a3b EH and a 36 Ea. g= (926) (936) (46) = 92 936 ab & H Not nomel. = * Ho * Hg + index in d oo nomel. index in 2 so nomel. it as also homel. Hence it will contain all the element. + o and Hio are 20, H1, H2, H3, Ho, Hg. Donnel subroups of Dy. Now, we have to find factor group of Dy up to somosphišim . (cleuletting for non trivial subegoloupal). Note: Of His pomal cubi qoroup of C. Define alu = Ho lafay then set etut together reith the binocy composition defined by Halb = Hab is velled The identity elt of als in H. and og are of order 4 19. Hayo 20= 2 they factor group H2, Ho, So, au e a u Z2 H₂ a u z 2 He a À Z2 Ha

and H₃ is H3 is of onder 2 le H₃ = {e, a 27 but the order of sque of any non-trivi a element le az (92)2 is identity That is, the squel of any element of H₂ ie in identity conet &e, 9²% of <92) which means every G his order that elt of 2 ... 30, G 29²5 a H₃ Z 2X 22 Kleins 4-group. That is the factor qroups of Dy. Ол Z, and Z2 X 22

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