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Three finite groups with the same numbers of elements of each order

Suppose that we have two finite groups $G$ and $H$ such that for each $n\in\mathbb{N}$ $\left|\left\{ g\in G:\text{ord}\left(g\right)=n\right\} \right|=\left|\left\{ h\in H:\text{ord}\left(h\right)=n\right\} \right|$ meaning, the number of elements of order $n$ in each group is equal. Does that imply that they are isomorphic? (With abelian groups I think it is true and easily deduced from the fundamental theorem of finitely generated abelian groups)

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    Interesting example. Thank you.2012-08-12

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