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Show that a group of order $5$ is always abelian.

I know that if the binary operation $*$ is defined on a group $G$ is commutative (i.e. $a*b=b*a\ \forall a,b\in G$),\ then G is called a commutative(or abelian) group. But I don't know how to solve a group of order $5$ is abelian. Thanks in advance!

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    How much group theory do you know? If you know Lagrange's Theorem, this is pretty easy. If you are only beginning to learn about groups, then this can be rather tedious.2012-04-12
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    i know some result of group theory and i know that every group of prime order is cyclic and every cyclic group is abelian that's why group of order $5$ is abelian. And this things only give me validity of given statement but i want to prove this by definition of abelian group and order of an element of a group2012-04-12
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    Sorry, but that makes no sense to me. If you know that every group of prime order is cyclic, and you know every cyclic group is abelian, then why do you think this is not a proof? The implication that "every cyclic group is abelian" *uses* the "definition of abelian group".2012-04-12

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