Hello,

I have a question about the CycleIndexPolynomial command contained in the Group Theory package. The answers it gives for computing the cycle index of the dihedral group and symmetric group appear to has a missing term in its answer. If my knowledge of how the cycle index is computed, it appears to be missing a term that occurs when cycles of length greater than one is combined with terms with cycles of length one. For example, here is what I get when I use Maple 18 on my Windows 8.1 machine to compute the cycle index of the following dihedral groups:

> with(GroupTheory):

> CycleIndexPolynomial(DihedralGroup(3), [x || (1 .. 3)]);

> CycleIndexPolynomial(DihedralGroup(4), [x || (1 .. 4)]);

> CycleIndexPolynomial(DihedralGroup(5), [x || (1 .. 5)]);

 Now if you compare these answers from the cycle index I found at the link:

 http://mathworld.wolfram.com/DihedralGroup.html

 you will see the answers is given as

  =

 =

 =

The disagreement occurs at the terms that contain representations of cycles of length two x2 multiplied to the one-cycle representation term x1.

The same issue happens with the symmetric group. Here is what I get for the following Maple commands:

> with(GroupTheory):

> CycleIndexPolynomial(SymmetricGroup(3), [x || (1 .. 3)]);

> CycleIndexPolynomial(SymmetricGroup(4), [x || (1 .. 4)]);

> CycleIndexPolynomial(SymmetricGroup(5), [x || (1 .. 5)]);

However, as can be seen at the link

http://mathworld.wolfram.com/SymmetricGroup.html

these answers are not in agreement with these:

  =

 =

 =

Again, the difference seems to be with terms combined with the one cycle term representation x1.

Is there something I am not interpreting correctly? Thank you for your help.

Neil Sigmon