Order of elements in finite group

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Homework Statement



If G is a finite group such that all elements have prime order(other than the identity). If G has a non-trivial center then show every element has the same order.

Homework Equations





The Attempt at a Solution



Since we know that G has a center, we know that there is atleast one element that commutes with every element in G. I'm not sure how we can show that all elements have the same order from just this.
 

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