# The $h-$th power of every element in a finite group of order $h$ is the identity element of the group

I work in population dynamics. I want to show somewhere in my work that the $h-$th power of every element in a finite group of order $h$ is the identity element of the group. I guess this is elementary result but it would be nice if somebody show me how to do it or where to find it. Thanks

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## 1 Answer

The order of every element of a group is a divisor of $h$ by Lagrange's Theorem. So if $x$ is an element of your group of order $k$ then $h = k\cdot l$ for some $l.$ Hence we have

$$x^h = x^{k\cdot l} = (x^k)^l = (e)^l = e.$$

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Can you provide a name of a book where I read this result for citation purpose. –  Zizo Nov 29 '12 at 7:35
@zizo abstract algebra by dummit and foote –  jim Nov 29 '12 at 7:40
@jim I can't open this book from google :-(. –  Zizo Nov 29 '12 at 7:44
Why exactly do you need to cite this theorem? One reason a theorem has a name is so that you can reference to it :) –  Jernej Nov 29 '12 at 7:47
Okay, thanks. I am master's student, I didn't know this. –  Zizo Nov 29 '12 at 7:48