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I am having trouble seeing why a finite divisible group is necessarily trivial. Why does this have to be the case?

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Can someone explain me what "a finite divisible group" does mean? Thanks. – Marc Aug 8 '14 at 14:14
up vote 7 down vote accepted

Let $n=|G|$. If $g\in G$, then by divisibility there exists $h\in G$ with $h^n=g$. But $h^n=1$.

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Assume $G$ is a finite divisible group and let $g\in G$, there is some $g'\in G$ s.t $g'^{k}=g$ for all $k\in\mathbb{N}$ and for $k=|G|$ we get $g'^{|G|}=g$ thus $g=e$ and $G=\{e\}$

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