When will two groups be isomorphic???

Question

If two finite groups G and H have same element structure i.e. they both have same number of elements of every particular order,then will they be isomorphic to each other??If not,give some counter example. And what is the intuition behind having same element structure..?

Answer 1

No, that is not enough. The elementary abelian group $\;C_p\times C_p\times C_p\;$ of order $\;p^3\;$ and the Heisenberg group

$$H:=\left\{\;\;\begin{pmatrix} 1&a&b\\0&1&c\\0&0&1\end{pmatrix}\;/\;\;a,b,c\in\Bbb F_p\;\;\right\}$$

have both the same number of elements of each order (only order $\;p\;$ and $\;1\;$), yet one is abelian and the other one isn't.