Question

(5 points) Recall the Definition: A subgroup H of G is called a normal subgroup of G if gH = Hg for all g E G. If so, we write H G. Mark each of the following true T or false F (using the CAPITAL LETTER T or F. Recall that if a statment is not necessarily ALWAYS true, then it is false. - T ח 1. Every subgroup of (Zn, e) is normal. 2. The cyclic group (f) is a normal subgroup of D3. 3. The cyclic group (r) is a normal subgroup of D3 . 4. If gHg-1 CH for some g E G, then H GG. 5. If H G , then |G: H = 2. 6. If G:H = 2, then H = gHg-1 for all ge G. 7. Every subgroup of Sn is normal. 8. The trivial group {e} is a normal subgroup of every group G. 9. If gHg-1 C H for all ge G, then H G. ד ד

Answer 1

T then l. since (Zn ) is bylic Group of order nas abelion + Every subgroup of abelion Group is normal so (true} do ip fake f= 1, 63 (1):71 G= Og zs 0(4)=6 D3 has subgroups of .. order ", 21316 if <*>=H + 0(H) = 3 3 H is normal also. its Normal Subgrups of pn. nisodel it <Roy, aln Here n=6 d=1121316 < R'), {R?), <R3), TR6) Normal Re Reflection. Rerfection H . n is even then <R²,,F7,2828f7 is normal in on to ANATILIPART f7 not normal as n=3 TE But xry is normal in D3 2 6 pase , ③ True. ① gHgleH for some gEG # Hoy (false) 6 take G= 216 H = 2/2 , so HAG But 19:41= 3 (false) 6 19:11=2 zo HAG & gH= 49 kg R eme :) H= g "Hg (True)

Ô @ take Soir Sq H= S3 as so has normal subgroup {e} , {Ad}, ki so false ses is Normal in Any Group - (By Definition) .. . True касне 4 7 НАС ън с 3 + H2 SH от: H1 (True) в)