Question

Every cyclic group is a/an ______

(a) infinite subgroup

(b) abelian group

(c) monoid

(d) commutative semigroup

The question was asked in semester exam.

I'd like to ask this question from Cyclic Groups topic in portion Groups of Discrete Mathematics

Answer 1

Correct answer is (b) abelian group

Explanation: Let C be a cyclic group with a generator g∈C. Namely, we have G={g.Let x and y be arbitrary elements in C. Then, there exists n, m∈Z such that x=gn and y=gm. It follows that x*y = gn*gm = gn+m = gm*gn = yx. Hence, we find that xy=yx for any x,y belongs to G.Thus, G is an abelian group.