Please ASAP

(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. TABLE 1. Table for V. * POP1 P2 P3 M1 M2 81 82 Po PO P1 P2 P3 Mi M2 81 82 pi pi P2 P3 po di 82 M2 M P2 P2 P3 Popi M2 Mi 02 si P3 P3 Po Pi P2 02 01 Mi M2 MM 82 M2 di Po P2 P3 P1 M2 M2 81 Mi 02 P2 Po P1 P3 81 di Mi 82 M2 P1 P3 po P2 82 82 M2 81 Mi P3 P1 P2 Po Subgroups of D4 P3 P3 Po P1 P2 02 01 Mi 12 P2 P3 M2 M2 di M 82|p2|PO|PL di di Mid2|M2|p1|p3|po|p2 da 82 M2 di MP3 P1 P2 PO Subgroups of D4 {po, P2, M1, M2}, {P0, P2, P1P3}, {P0, P2,01,02} {Po, P2}, {po, , M1}, {Po, M2}, {po, 81},{Po, 82} {po} (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. TABLE 1. Table for V. * POP1 P2 P3 M1 M2 81 82 Po PO P1 P2 P3 Mi M2 81 82 pi pi P2 P3 po di 82 M2 M P2 P2 P3 Popi M2 Mi 02 si P3 P3 Po Pi P2 02 01 Mi M2 MM 82 M2 di Po P2 P3 P1 M2 M2 81 Mi 02 P2 Po P1 P3 81 di Mi 82 M2 P1 P3 po P2 82 82 M2 81 Mi P3 P1 P2 Po Subgroups of D4 P3 P3 Po P1 P2 02 01 Mi 12 P2 P3 M2 M2 di M 82|p2|PO|PL di di Mid2|M2|p1|p3|po|p2 da 82 M2 di MP3 P1 P2 PO Subgroups of D4 {po, P2, M1, M2}, {P0, P2, P1P3}, {P0, P2,01,02} {Po, P2}, {po, , M1}, {Po, M2}, {po, 81},{Po, 82} {po}