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16. Up to isomorphism, the only infinite eyelic group is Z, under the usual addition. What are the subgroups of Z? Establish the isomorphism between Z and 22. Establish the isomorphism between Z and 3Z. In general, between Z and nz for n a positive integer. 17. According to the Fundamental Theorem of Finite Abelian Groups, up to isomorphism, a finite abelian group of order n is isomorphic to a direct product of cyclic groups of prime power order and the de composition is unique up to the order in which the factors are written. List the finite abelian groups of order: 5, 6, 10, 12, 16, 20, 36, 100, 180

Answer 1

② we know that z is an infinite cyclic group So the subgroups of z are also cyclic, because, Subgroup of a cyclic group is cyclic Subgroup of z generated by <1> . {-1,-1, 0, 1, 2, .... - 2 C2>= (-4,-2.0, 2, 4, ....)-22 ens = {-an, -n, o, nian...] = n2 Subgroups of z are nz, where nez . det cleary fiz→ 22 defined by f(x) = 2x fis a homomorphism, feety) = 2(x+y) - 2x + ay f is one-one, f(x) = f (y) =2xQy x=y for any yea2 , %E z and f(2)=2.4 = y f(x) = ax z dedines on isomorphism between and az.

1 In similar way O f (60) = 3x defines an isomorphism between 2 and 32. In general, if n is any positive integer fi z nz , datined by f(x) = n* The map is well detined. w f - homomorphism, ffety) nec+y) net ny - f() + f(y) 6) f is one-one, fasa fry > nesny -> x=y Enemy. (ii). f is onto, for any yenz, y ez O and fx) = sinay, - f: zanz detined by f(x) = nx. is an isomorphism

15 2,2-2x 2.3 . 2.28 29 2,1 2,7 25х2. zu x 24 220 2ax 22 x 25 го - 36 - 236 24x2-4

100 - Zioo Zas* Zax Za Zgx Zs x Z4 zs + Z5 x Zzxza Zigo 180 Zax 22 x Zq x 2s Zax Zax Zzx 2; XZs 24x Zzx Z3X Z5