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Let G and H be finite groups of the same order. If for any positive integer n, the number of elements in G having order n is the same as that in H, then is it true that G and H are isomorphic?

shansh0201
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  • There are 34 different groups of order $p^6$ and exponent $p$ for all primes $p\geq 7$. This is the first result that came up in a Google search. You should play with GAP to find examples of these things —I am sure there are many much smaller ones. – Mariano Suárez-Álvarez Nov 13 '16 at 09:10
  • You can also look in the list of all groups of order 81 for those of exponent 3: there are several. See http://groupprops.subwiki.org/wiki/Groups_of_order_81 – Mariano Suárez-Álvarez Nov 13 '16 at 09:16
  • The morale of the story is «examples, examples, examples!» Lots of people have put an immense amount of effort in things like GAP and the groupprops site for us: take advantage of them when thinking about any problem. – Mariano Suárez-Álvarez Nov 13 '16 at 09:18
  • The answer seems to be no. This question provides an example.https://math.stackexchange.com/questions/693163/groups-with-same-number-of-elements-of-each-order – Tyler Nov 13 '16 at 09:29

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