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prove that if G is a finite and abelian group and m is the least common multiple of the order of it's element, that there is an element of order m.

My idea:

if ai are the elements of G, the order of a1*a2 is lcm(a1,a2) and the result follows directly when applied to all ai... but why is this correct and why is this only for abelian groups?

My idea:

if ai are the elements of G, the order of a1*a2 is lcm(a1,a2) and the result follows directly when applied to all ai... but why is this correct and why is this only for abelian groups?

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