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Groups vs. permutation groups
The cycle index is a property of a permutation group, not of a group. The permutation group article rightly draws a distinction between these notions; it points out that two groups G and H can be isomorphic as groups without being isomorphic as permutation groups. For example, consider the group G generated by the single permutation (1 2 3 4 5 6), and the group H generated by the permutations (1 2) and (3 4 5). G and H are isomorphic as groups—they are both isomorphic to C6—but they are different permutation groups, and they have different cycle indices. But this article glosses over or ignores this important point in many places.
For example, the article says "The cyclic group C6 contains the six permutations … and its cycle index is …". We can think of G and H of the previous paragraph as corresponding to different homomorphisms from C6 to S6. Here the "G" embedding of C6 has been chosen, and the "H" embedding ignored, but the article could just have easily have gone the other way. Is it correct to speak of "The cyclic group C6" in this case? Don't algebraists have some more precise terminology that they would use here? Would they talk explicitly about the homomorphisms?
I do not have enough expertise or command of the technical terminology to repair this myself, so I am going to tag this article as needing attention from an expert. —Mark Dominus (talk) 16:27, 10 February 2012 (UTC)
- Tweaked two spots that I thought strayed furthest from the permutation group assumption. Rschwieb (talk) 13:01, 11 February 2012 (UTC)
I'm still concerned about the phrase I quoted above, "The cyclic group C6 contains the six permutations", and others like it. Shouldn't this be phrased in terms of a representation of C6 as a permutation group, or in terms of a homomorphism from C6 to S6? —Mark Dominus (talk) 14:51, 14 February 2012 (UTC)