Question

Prove that an element has an order in an abelian group

Answer 1

2down votefavoriteI will quote a question from my textbook, to prevent misinterpretation:Let GG be a finite abelian group and let mm be the least common multiple of the orders of its elements. Prove that GG contains an element of order mm.I figured that, if |G|=n|G|=n, then I should interpret the part with the least common multiple as lcm(|x1|,…,|xn|)=mlcm(|x1|,…,|xn|)=m, where xi∈Gxi∈G for 0≤i≤n0≤i≤n, thus, for all such xixi, ∃ai∈N∃ai∈N such that m=|xi|aim=|xi|ai. I guess I should use the fact that |xi||xi| divides |G||G|, so ∃k∈N∃k∈N such that |G|=k|xi||G|=k|xi| for all xi∈Gxi∈G. I'm not really sure how to go from here, in particular how I should use the fact that GG is abelian.