Question

(7)(20 points) (a) State the Fundamental Theorem of Finitely Generated Abelian Groups. (b) List all abelian groups of order 2450 up to isomorphism. (c) Show every abelian group of order 2450 has an element of order 70.

Answer 1

Vo 7:20 AM 0.3KB/s na 44 or each positive integer n, let Zn =Z/nZ be the cyclic group of order n. Theorem (Fundamental Theorem of Finitely Generated Abelian Groups) Theorem. Let G be a finitely generated abelian group. Then it decomposes as follows: G - Z" x Znx Zin, X xZn, 9 for some integers r, ni,n2,...,ns satisfying the following conditions: 1. r > 0 and ni > 2 for all i, and 2. Ni+1|ni for 1 <i< s - 1. The decomposition of G satisfying these conditions is unique. The integer p in the decomposition (*) is

Kevin James Fundamental Theorem of Finitely Generated Abelian Groups DEFINITION O A group G is finitely generated if there is a finite subset A C G such that G =< A>. @ For each 0 <re Z, let Z' = Z x ... x Z be the direct product of r copies of Z, where we take Z° = 1. The group Z' is a free abelian group of rank r. THEOREM (FUNDAMENTAL THEOREM OF FINITELY GENERATED ABELIAN GROUPS) Let G be a finitely generated Abelian group. Then, O G = Z' x Zni X ...x Zng for some integers r, N1, ..., ng satisfying the following conditions. 1 r> 0 and n; > 2 for 1 <j <s, and 2 ni+1|n; for 1 <i< (s – 1). 2 The expression above is unique. Kevin James Fundamental Theorem of Finitely Generated Abelian Groups DEFINITION The integer r in the above Theorem is called the free rank or Betti number of G and the integers n; are called the invariant factors of G. The description of G in the Theorem is called the invariant factor decomposition of G. nae Fundamental Theorem of Finitely Generated Abelian Groups Kevin James

0 abelian Thim Fundamental Theorm Koitely Generaled Abelian Groups statement der a be a fortely generated. group (1) There is a Unique na nega fire integer s such that the number of infinite cyclic cyclic sunmands Pn Ciny de composition of G as a direct Som of Cyclic group is precisely so (i) either a is hee abilian or there is a unique list of (not neccecooily dos hinct) positive integers me, M2,- .omt such that m,> 1) Milmal..In and G q tm, Zm2 0 0 2m. Af with f bee abdian. Ces either a is bee abelian or there Ps list of positive integers pist, p.22 I Pask which is unique except. for the order of its members such That Pl. P2) :: Dk are (not becelicon- aly distinct) primes, S1,S2, Stare (not necesarily distinct positive Gå ZP154 Zp252 © 2006 ZP** with f free ablican integers and

2450 i we need to 2 5.1225 51 245 대 7 그 (6) Cilven orden Sol To find all abelian group (upto isomorphism of order 2450 sol 181 21 2450 factor 2450-2 2450= 2 ².52 72 49 2450 72.5²2 6 Now for the factor 2 Yiels 22. (i) Now for the factor :5², we get the following following non- Osormorphic groups o Z25 and 25 0 5 (1) Now for the factor 72 yields Zluq and 770 77 gel Zz Ø 22 25 Z25 6 2/49 2/2 ① 2 5 0 2 ④ Zua Zz o 225 2774 “) 220 Z 2 0 2 0 25 ④ 24 ④ 27 so we ged a total of 4 possible of order 9450 groups So we (0) (3) 3

with ) B z 2252 (This sheere 7 each grouph factors into A BOC | Alz 49, IB = 25 (6/21 The Abelian groups A coresponds to poothons of B comesponds to paslihons of 5 c corresponds to partition of a Thus we hav А *772 or Zi21 or 2s ② 25 C = 212 So there are are 2.2-1 = 4 Abelian. 2450, up to Isomor- direct sums A 0 B 0 C. es There are 4 Abelian groups onden 9450, uplo Psomorphismos (C) sol The Abelian groups of order 2450 arif, Up to Psomorphism, 496 Group Orden I ll element of Dades to 224 225 22:49 017) 2222225+Z177 (0,52) 3 224 25 + 25 Azuq (1,4,7) 4 ₂025 4 24+247 & (1,1,1) We have to take Lim of all. gooups of orden phism. - conclusion ） ) (1,5,7) D 2

Schas an element of order to fom (1) Z20250240 2 *55 7X7 & (1) 65(5) 6767 5-7) (1,5,7) o 70(1,5,7) for Z10at we write it as 27+/ so sto has only 1 Seren 72071567.97492 + 507x7 76.5.4 (),1,1) = to (1,7) 21272/25 €24.774 20) 5(5) -711) »5.7 (1,5;)) 4) 72 75+775 67+U12 2:5-7z2(1)-510 2 • 5X 5 7. N 70 (1,1,1) (1) Conclusion So frecry abelian group of 2450 has an element of order to conclusion GIVE THUAMBIO P PLEASE iDO O Prote