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Which of the following is false? fa) any abelian group of order 27 is cyclic (b) any abelian group of order 14 is cyelic c) any abelian group of order 21 is cyc
Which of the following is false? fa) Any abelian group of order 27 is cyclic (b) Any abelian group of order 14 is cyelic c) Any abelian group of order 21 is cyclic (d) Any abelian group of order 30 is cyclic
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Anuj s. Best Answer
(b), (c), (d) are true. By the main theorem on finite abelian groups, they are direct products of cyclic groups, necessarily of orders 2 and 7 (b), 3 and 7 (c), and 2, 3, 5 (d). Direct products of cyclic groups of coprime order are cyclic.
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Anuj s.
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a) is false, because 27 is divided by 9, which is 3 * 3. For any number that is divided by the square of a prime, there is an abelian group that is not cyclic. In this case, the groups Z3 x Z3 x Z3 and Z3 x Z9 are the two non-cyclic abelian groups of order 27.
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