Problem 5 15 points Let G be a finite abelian group and supp

Problem 5 (15 points). Let G be a finite abelian group, and suppose that |G| = m1... mn where m1,..., mn are pairwise relatively prime integers, that is

Assume also that there exist elements x1, . . . , xn ? G such that |xi| = mi for every1 ? i ? n. Prove that G is cyclic.

Solution

Here |G|=m1m2...mn and there exist elements xi such that |xi|=mi, 1<=i,j<=n and i not= j.

so the element x1x2...xn has order m1m2...mn=|G|.

i.e. <x1x2...xn>=G

Hence G is cyclic.