Question

(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

Fundamental theoren of thearen of finitely gene abelian groups finitely genarated abelian group following Let Gbe a Then G²10x7,x ... XEn Son some integers 8, s. 82, ..Se as Satisfying the Conditions Code po and dj 2 Ser saj < 3 ② Nitilai for 1sis (S-1) and the expression above es unique 1 2450 = 2 x 5² x 72 powes (1, 2, 2) (1, 2, 2) = (1, 2, 2) (1, 2, 1+1) = (1, 1+1, 1+1) nou n (1, 1+1, 2)

Abelian groups are o 2/2 725 49 2) 727.52 7 5) 21275Z 7777 u 212 7/ s71749 70- 2x5X7 Sinire group abelian Cauchy Theorem - Let G be a and plo(G) where pesa Prime number then there is an elenet a ze EG such that o(a)=P e is the édentity elements of the group row (G) = 2450 and 212450 Since 2 is Prime by Cauchy theoren there has an element of Onder 2 say ocas=2 Similonly 5/2450 hence there is an b of onders 0 (6)=s row (2,5) = 1 (god is 1) ocab) = 10 hence there is an elenet of order 10 elemet

theoren there has 7/2450 then an element & of order 7 (0) = o cab) = 10 again by caucly group es Since ged (7, 10) = 1 and the Sinite abelion group olabe)=70 herce group G of ander 2450 there hasan element of onder 70. Thenn you 0 in the