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Theory: If G is a finite abelian group, p is prime and p divides the order of G then G has an element of order p.

Can anyone think of a counter example for a number n that is not prime, divides the order of some group, and the result proves its false.

I have looked at cyclic groups, the group of nth roots of unity and every finite abelian group I know, and no matter what, if some number divides the order of a group it has some element in it that has an order of that number. It doesn't really matter if it's prime or not. Any examples?

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The Klein group is abelian and order four, but no element has order four. More generally, a group that is the direct product of two cyclic groups that have orders that are not coprime will not have an element of the order of the group.

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what would be the order of any element in $\Bbb Z_8^*=\{1,3,5,7\}$, so there is no element of order $4$

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Consider the direct product of, say, four cyclic groups of order 3, which has order 81 and is abelian. Does it have elements of order $9$, or 27?

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