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

Generated (a) statement of the fundamental Theoreu of Fini Abelian Groupe : initely Let G be a finitely generated abelian group. Then G can be decomposes Gr 7 x 7m x Zx ... x Zmx → (!) for some intigere enn and these integers conditions : satisfies the following i> x 70 and na 7,2 for all i= 1, 2, ..,k. ü nir, Ini for Isisk-1, The expression (1) along with the above too conditions is unique. (6) We have 2450 = 2x5²x7? Therefore, the list of all abolian groups of order 2450 upto isomorphism is as follows: Zx 225 x Zz x 77 Zq X Z XZ, X Zyx Zy divides ni is ni niti x7 2 x Z25 -49 х Z *Z* Zag Z2 5 every Since in the expression 2450 = 2x57x72 all the primes 2,5 and 7; then abelian group of order 2450 must has an element of order 2x 5x7 = 70 are