(Again) Number of rotation subgroups of S_n

simone at cs.york.ac.uk simone at cs.york.ac.uk
Tue Nov 15 17:31:12 CET 2005

Dear seqfans,

On the 10/10/2005 Neil Sloane posted the following message:

>A friend suggested the following set of sequences

>Let S_n be the symmetric group of all perms of n letters,
>of order n!

>A transposition is a permutation like (3,5).
>A reflection is any element of order 2, that is, a product of disjoint
>A "rotation" is a product of two transpositions, not necessarily
disjoint, >so either (1,2,3) or (1,2)(3,4), etc.

>How many subgroups of S_n have the property that they are
>generated by rotations?
>How many non-isomorphic ones?

>How many subgroups of S_n have the property that they are
>generated by reflections?  (Is this the Bell numbers, A000110 ?)
>How many non-isomorphic ones?  (Again, this may be a well-known sequence)

>Same two questions for groups generated by rotations AND reflections?


Now, Peter Cameron has written the attached working notes. However, the
problem is not solved yet, and I am working on it.

Is there anybody who has time to help me?

Thanks and best regards,

Simone Severini
Department of Mathematics and Department of Computer Science
University of York, YO10 5DD York, U.K.
Tel: +44 (0)1904 433072
Web: www-users.york.ac.uk/~ss54
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