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Commutative Group (Posted on 2006-09-18) Difficulty: 2 of 5
Here is a simple problem from abstract algebra.

Prove that a group with exactly five elements is commutative.

See The Solution Submitted by Bractals    
Rating: 3.0000 (1 votes)

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Lagrange Argument | Comment 5 of 8 |
The order of an element must divide the order of the group by Lagrange's Theorem. Since the group has prime order, all elements have either order 1 or 5. Since the identity element is the unique element with order 1, the remaining 4 elements must all have order 5. Any of these elements is thus a generator of the entire group, showing that the group is cyclic. Cyclic implies abelian, completing the proof.
  Posted by Jason on 2006-09-18 14:09:27
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