Question

1. Let G = U(27) and let H = {1, 26}. Find all left cosets of H in G. 2. Let G = Z50 and consider it's subgroup H = (5). Find all coset representatives of 3 + H.

hello guys , help me please to solve this
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( abstract algebra )

Answer 1

1. let G=U (27) = {1,2,4,5,7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 200, 25, 26} let H = { 1,263 H = £1126} 2H2= £225} 10 H={10, 17} 11H=211 11H={11,16} 13H={13, 14} 14H={144, 13} 4H = {4.222 SH={5, 22} 8H = 28, 19} 16H = £16, 117 ITH={17, 10} 19H = 419,8} 20H = 220,7} 22H = {22,5} 23H = {23,4} 25H = {25,2} 26H = {2611} . left cosets of H. This all are

G22150 H=<s> -LO,S, 10,15, 20,25,30 35, 40,45} all coset represenetitives of To find 3+Hi A = 23,8, 13, 18, 23,28 33, 38,43,483 as ath qtH = b th its a-btH = H iff I a-bEH. so to Hind b site 3-6b EH. l-eb=8,13,18,23,28,33,38,43,48 lie. 3tH=8+H= 18+H =234 H 2 =28tH=38tH=33tH=48+4=48+H This all are coset representatives of 34H Thank you.