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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..?


marked as duplicate by Dietrich Burde, user228113, Derek Holt group-theory Jan 16 '17 at 11:42

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  • $\begingroup$ The answer is yes for abelian groups, but no in general. $\endgroup$ – Arthur Jan 16 '17 at 10:09
  • $\begingroup$ can you give some counter example for non-abelian groups..?? $\endgroup$ – Sumit Mittal Jan 16 '17 at 10:11
  • $\begingroup$ @Arthur -- Do you know of an example of two finite (non-abelian) groups which have the have the same number of elements of each order, and yet are not isomorphic? $\endgroup$ – quasi Jan 16 '17 at 10:12
  • $\begingroup$ @Sumit Mittal -- We asked the same question at the same time!, $\endgroup$ – quasi Jan 16 '17 at 10:14
  • $\begingroup$ @quasi This MO question, for instance, has some examples. $\endgroup$ – Arthur Jan 16 '17 at 10:16

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.

  • 1
    $\begingroup$ Nice example -- thanks. $\endgroup$ – quasi Jan 16 '17 at 10:16

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