(a) µn, i.e. the complex n-th roots of unity under multiplication: This group is cyclic. The complex n-th roots of unity are the solutions to the equation z^n = 1, where n is a positive integer. These roots can be expressed in the form cos(2πk/n) + isin(2πk/n), where k is an integer from 0 to n-1. The number e^(2πi/n) is a generator of this group, because when it is raised to the power k (for k from 0 to n-1), it generates all the n-th roots of unity. Therefore, the group is cyclic.

(b) The group of symmetries of a rectangle: This group is not cyclic. The group of symmetries of a rectangle includes four rotations (0, 90, 180, and 270 degrees) and four reflections (about the horizontal axis, the vertical axis, and the two diagonals). No single symmetry can generate all the others through repeated application, so the group is not cyclic.

(c) The rational numbers Q under addition: This group is not cyclic. A cyclic group is one that can be generated by a single element. However, no matter which rational number you choose, you can never generate all rational numbers by repeatedly adding it to itself. Therefore, the group of rational numbers under addition is not cyclic.

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(a) µn, i.e. the complex n-th roots of unity under multiplication: This group is cyclic. The complex n-th roots of unity are the solutions to the equation z^n = 1, where n is a positive integer. These roots can be expressed in the form cos(2πk/n) + isin(2πk/n), where k is an integer from 0 to n-1. The number e^(2πi/n) is a generator of this group, because when it is raised to the power k (for k from 0 to n-1), it generates all the n-th roots of unity. Therefore, the group is cyclic.

(b) The group of symmetries of a rectangle: This group is not cyclic. The group of symmetries of a rectangle includes four rotations (0, 90, 180, and 270 degrees) and four reflections (about the horizontal axis, the vertical axis, and the two diagonals). No single symmetry can generate all the others through repeated application, so the group is not cyclic.

(c) The rational numbers Q under addition: This group is not cyclic. A cyclic group is one that can be generated by a single element. However, no matter which rational number you choose, you can never generate all rational numbers by repeatedly adding it to itself. Therefore, the group of rational numbers under addition is not cyclic.

Knowee AI · Accepted Answer

(a) µn, i.e. the complex n-th roots of unity under multiplication: This group is cyclic. The complex n-th roots of unity are the solutions to the equation z^n = 1, where n is a positive integer. These roots can be expressed in the form cos(2πk/n) + isin(2πk/n), where k is an integer from 0 to n-1. The number e^(2πi/n) is a generator of this group, because when it is raised to the power k (for k from 0 to n-1), it generates all the n-th roots of unity. Therefore, the group is cyclic.

(b) The group of symmetries of a rectangle: This group is not cyclic. The group of symmetries of a rectangle includes four rotations (0, 90, 180, and 270 degrees) and four reflections (about the horizontal axis, the vertical axis, and the two diagonals). No single symmetry can generate all the others through repeated application, so the group is not cyclic.

(c) The rational numbers Q under addition: This group is not cyclic. A cyclic group is one that can be generated by a single element. However, no matter which rational number you choose, you can never generate all rational numbers by repeatedly adding it to itself. Therefore, the group of rational numbers under addition is not cyclic.

Which of the following groups are cyclic? Explain your answers.(a) µn, i.e. the complex n-th roots of unity under multiplication;(b) the group of symmetries of a rectangle;(c) the rational numbers Q under addition

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