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.
Is cycle index unique for every distinct group up to isomorphism?
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group-theory
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0No...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 $3$. – 2012-05-21
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0Thanks 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? – 2012-05-21
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0I'm not sure about the second part, which is why I didn't post that as an answer! – 2012-05-21
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1Sorry... what is the definition of cycle index? – 2012-05-21
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1http://en.wikipedia.org/wiki/Cycle_index – 2012-05-21
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0"Cycle index" is not a function of a group; it's a function of a pair consisting of a group and a permutation representation of it, and it depends on both of these pieces of data. – 2015-10-18