Abstract Algebra (Direct Products of Groups)
Abstract Algebra (Direct Products of Groups) Let G1, G2 and H be finitely generated abelian groups....
2. Let G1, G2, and G3 be groups. Prove the following: a) If G1 = G2, then G2 = 61. b) If G = G2 and G2 = G3, then G =G3.
Abstract algebra A. Assume G is an abelian group. Let n > 0 be an integer. Prove that f(x) = ?" is a homomorphism from Got G. B. Assume G is an abelian group. Prove that f(x) = 2-1 is a homomorphism from Got G. C. For the (non-abelian) group S3, is f(x) = --! a homomorphism? Why?
Using the Fundamental Theorem of Finitely Generated Abelian Groups, classify the following factor groups. (a) Z x Z/ < (5,6) > (b) Z3 x Z6 / < (1,2) > (c) Z x Z/ < (4,4) >
(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.
(Abstract Algebra) Please write clearly 1. Abelian Groups. [Purpose: Apply prior concepts in a new context.] Prove that if G and H are Abelian groups, then Gx H is an Abelian group. 1. Abelian Groups. [Purpose: Apply prior concepts in a new context.] Prove that if G and H are Abelian groups, then Gx H is an Abelian group.
(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.
(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.
Use the Fundamental Theorem of Finitely Generated Abelian Groups to answer the following: a. Find all abelian groups, up to isomorphism, of order p3 where p is a prime b. Use part (a) with a suitable p to list all possible abelian groups that are isomorphic to (Z2x From this list, identify the abelian group that is isomorphic to (Z2xZ8)/(1, 4))
ASAP (1) (20 points) Using the Fundamental Theorem of Finitely Generated Abelian Groups, classify the following factor groups. (a) Z x Z/ < (5,6) > (b) Z3 x Z6/ < (1, 2) > (c) Zx Z/ < (4,4) >
8. (20 points) Let G Zs x Zg and let H be the cyclic subgroup generated by (3, 3). (a) Find the order of H (b) Find the orders of g = (1,1) + H, h = (1,0) + H and k = (0,1) + H in G/H (c) Classify the factor group G/H according to the fundamental theorem of finitely generated abelian groups. 8. (20 points) Let G Zs x Zg and let H be the cyclic subgroup generated...