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Are two groups isomorphic iff their cycle index is the same? Note that for every group there exists a permutation group to which it is isomorphic.

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No...because $\{1, (123), (132)\}$ has cycle index $\frac{1}{3}(a_1^3+2a_3)$ while $\{1, (123)(456), (132)(465)\}$ has cycle index $\frac{1}{3}(a_1^3+2a_6)$. These groups are isomorphic as they are both cyclic of order $3$. –  user1729 May 21 '12 at 10:01
    
Thanks for the contradiction example. But are there any normalized forms of the groups? A systematic way we could rewrite the later group into the first one? Perhaps my 2nd part of the original question is, are there any 2 groups that are not isomorphic and have the same cycle index? –  David Toth May 21 '12 at 12:31
    
I'm not sure about the second part, which is why I didn't post that as an answer! –  user1729 May 21 '12 at 13:06
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Sorry... what is the definition of cycle index? –  Arturo Magidin May 21 '12 at 16:32
    
en.wikipedia.org/wiki/Cycle_index –  David Toth May 21 '12 at 19:29
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