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Abstract / Modern ALG: Ker (PHI) and phi (-3.2) for phi: ZXZ-> Z, where phi (1.0) = 3 and phi (0,1) =- 5. Calculate …?

Calculate amounts of homomorphism. help!

If G and H are abelian groups and phi: G – H is a group homomorphism, then for all G1, G2, in G and M and N integers Z, we have phi (n g1 + g2 m) = n phi (G1) + phi m (G2). [This follows directly from the definition of group homomorphism] Now in Z x Z, we have (-3.2) = -3 (1.0) + 2 (0.1). Thus phi (-3.2) = -3 phi (1.0) + 2 phi (0.1) = -3 (3) + 2 (-5) = -19. Now, to calculate the kernel of a homomorphism phi: G -> H, it is necessary to determine exactly which elements g satisfying phi (g) = 0 in H. Now in your example, if (x, y) in Z x Z, then (x, y) = x (1,0) + Y (0,1), so phi (x, y) = 3x – 5a. Now phi (x, y) = 0 if and only if 3x-5y = 0. Since both x and y are integers, we have ker (PHI) = <(5,3)>.

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