(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
>transpositions
>A "rotation" is a product of two transpositions, not necessarily
disjoint, >so either (1,2,3) or (1,2)(3,4), etc.
>Questions:
>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?
>NJAS
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
--
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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