Let Ø:Z50 Z15 be a group homomorphism with Ø(x)=4x. Then, Ker(Ø)= O (0,15,30,45} O None of the choices O (0, 10, 20, 30, 40} O (0, 5, 15, 25, 35, 45}

Q: 5. Let G be the multiplicative group {[2], [4], [6], [8]} = Z10-

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Q: Give the lattice of subgroups of (Z2,), of (S,,0), and of (Z(20) ,)

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Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

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Q: 7. Consider an element z = the order of r is the least common multiple of the orders of its…

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Q: 14*. Find an explicit epimorphism from S4 onto a group of order 4. (In your work, identify the image…

A: A mapping f from G=S4 to G’ group of order 4 is called homomorphism if :

Q: Show that Z is not isomorphic to Zž by showing that the first group has an element of order 4 but…

A: The given groups: Z5x and Z8x That is, Z5x =Number of elements of a group of Z5 Z8x =Number of…

Q: 9. Classify the following groups in the sense of the Fundamental Theorem of Finitely Gener- ated…

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Q: Z, x is not a group.

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Q: 16* Find an explicit epimorphism from S5 onto a group of order 2

A: To construct an explicit homomorphism from S5 (the symmetric group on 5 symbols) which is onto the…

Q: A All B (ii), (iii) D E None (i), (iii) (i), (ii) Suppose such that (i): (ii): : Z→ Z40 is a group…

A: Please find attachment

Q: Let Ø: Z50 → Z15 be a group homomorphism with Ø(x) = 4x. Ø-1(4) None of the choices O {1, 16, 31,…

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Q: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y,    (x*y) = (y*x)   now associative…

Q: Let (G, *) be a group. Prove the following. (a) For each y = G, (y-¹)-¹ = y. (b) For x, y = G,…

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Q: 4. Prove that the set H = nEZ is a cyclic subgroup of the group GL(2, R).

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Q: 3. Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the…

A: Result: Let G, * be a group and H is a subset. The subset H is said to be subgroup, if for every a,…

Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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Q: 4. (a) (8 points) Prove the following two groups of order 8 are not isomorphic: Z8 Z4 × Z2.

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Q: Find all group homomorphisms from Z₂ XZ₁ to Z₂ x Z₂ xZ 3 4 2 3

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