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Let G be abelian, H and K subgroups of G with orders n, m. Then G has subgroup of order lcm(n,m)

I have gone through prrof here:- https://math.stackexchange.com/questions/465742/let-g-be-abelian-h-and-k-subgroups-of-orders-n-m-then-g-has-subgrou

But this uses assertion :-  if G is abelian and n divides |G| then G has a subgroup of order n.How is this assertion true. there are some existing examples where n divides G but there is no subgroup with that order.Please clear with explanation.

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