Let G be a group having a finite number of elements. Then For any a in G, there

Question

Let G be a group having a finite number of elements. Then For any a in G, there exist n in Z' such that a^n = e. for some n in Z^+, then G has n elements. Let b be the inverse of a in G. If there exists n in Z^+ such that a^n = e, then b ^n not equal e. If a is in G and a^n = e for some n in Z^+, then G is a cyclic group. Which of the following is true there exist a group in which the cancellation law holds. In every cyclic group every element is a generator. a cyclic group has a unique generator. Every group is a subgroup of itself. _______ Which of the following is a cyclic group The set of all integral multiples of 6 under addition. The set of 2 * 2 matrices having integer entries under addition. The set of symmetries of an equilateral triangle under function composition. The Klein 4-group. _________ Which of the following is false Every cyclic group is abelian. Not every abelian group is cyclic. Every group of order less than or equal to 4 is cyclic. If G is a cyclic group then G is isomorphic to Z under addition or to Z_s under addition modulo n where n is a positive integer. ______ Which of the following is false Every permutation is a one-to-one and onto function. The symmetric group S_3 is cyclic. The symmetric group S_10 has 10! Elements. Every subgroup of an abelian group is abelian.

Explanation / Answer

(8) D

(9) B

(10) A

(11) D

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