State whether each of the following statements is true or false. Then give an explanation, proof or counter example to justify your answer.    q1.Let g and h be elements of a group G such that gh = e. Then hg = e.  q2. For any two elements x and y in a group G, there is a unique element z in G such that y = xz.

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Q: 6. Show that for any two elements x, y of any group G, o(xy) = o(yx). %3D

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Q: 10. Let E = Q(V2, V5). What is the order of the group Gal(E/Q)? What is the order of Gal(Q(V10/Q)?

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Q: Q2: If G = R- {0} and a * b = 4ab ,show that (G,*) forms a commutative group? %3D

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Q: Suppose G is a group and r, be G so that r = b and r = b. Solve for a in terms of b.

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Q: 6. Prove the following groups are not cyclic : (a) Z x Z (b) Z6 × Z (c) (Q+, ·) (Here, Q+ = {q € Q…

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Q: any two elements x and y in a group G, there is a unique element z in G such that y = xz.

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Q: Define * on Q by a +b= qb Is Q a group under *? 210

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Q: 25. Prove that R* x R is a group under the operation defined by (a, b) * (c, d) = (ac, be + d).

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Q: (Identity) element for the group {Z, +} is 1. T

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Q: Which of the following groups are isomorphic? i) Z/75Z ii) Z/3Z ® Z/25Z iii) Z/15Z @ Z/5Z iv) Z/3Z @…

A: Groups (i) ℤ/75ℤ and (ii) ℤ/3ℤ⊕ℤ/25ℤ are isomorphic. Groups (iii) ℤ/15ℤ⊕ℤ/5ℤ and (iv) ℤ/3ℤ⊕ℤ/5ℤ⊕ℤ/5ℤ…

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Q: Which of the following is cyclic group а. R b. Z С. Q d. C

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