Question

Exercise 2.23. Suppose H and K are subgroups of G. Prove that HK is a subgroup of G if and only if HK = KH a abaža Exercise 2.24. Suppose H is a subgroup of G. Prove that HZ(G) is a subgroup of G. Exercise 2.25. (a) Give an example of a group G with subgroups H and K such that HUK is not a subgroup of G. (b) Suppose H, H., H. ... is an infinite collection of subgroups of a group G such that H, S H+ for all i E N. Prove that UH is a subgroup of G

Answer 1

Ano 2.23. det HK be a Supoose that e HK. As HK EHK.Ret h,k.Then khEKH.E HK nt KH, There fore HK KH.Similary, Kt tK and HK KH 17 Comvertsery lel HK= KH e e.e t HK,.Now let P,9E HK hk 1hak3 say. Thew pa(hk(hgk) h2(kzha)hg Alco,p k2) ptHKFron 1Reae Teanlts fo Uonr that HK kh KH HK. Thua pE HK det ab E Hz (a)) here a ete, be z G) H(z(6)) (an b Z) Aw 2 .24) ab - ba) Since ab ba Htela))(2(a)Hi (20)H CH(2(6), and A HG))- (26) ; (by (2(a)H abe (2la)) H Bnt ba E H(2(6)) io above reault 2- 23). the G a Corirder G(2,) and ts ubgr (22,4) and beo. 2.35 Ko (37,+). HUKinot H T ao CHUK CHUK t HUK but 3EK, 2EH. 3-2

Ano.2 a. 25)e eeUH: 2100 ,# 4. hnce eE H; Vi). Now lek a, b EUH 121 Then IN at H We hare 3 and for & me Caees subgraup of 6) ab 6 Hpc CH l abE He C H a CH TV Pab Ha H 121 and Com 1 &. H.AS