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..?
When will two groups be isomorphic???
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group-theory
group-isomorphism
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0The answer is yes for abelian groups, but no in general. – 2017-01-16
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0can you give some counter example for non-abelian groups..?? – 2017-01-16
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0@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? – 2017-01-16
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0@Sumit Mittal -- We asked the _same_ question at the _same_ time!, – 2017-01-16
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0@quasi [This MO question](http://mathoverflow.net/questions/39848/finite-groups-with-elements-of-the-same-order), for instance, has some examples. – 2017-01-16
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0@Arthur -- thanks for the link. – 2017-01-16
1 Answers
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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.
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1Nice example -- thanks. – 2017-01-16