Number of rotation subgroups of S_n

[Christian G.Bower]

------ Original Message ------
From: Jens Voss <jens at voss-ahrensburg.de>
To: seqfan <seqfan at ext.jussieu.fr>
Subject: Re: Number of rotation subgroups of S_n

>  > > 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?
>  >
>  > [...]
>  >
>  > I counted by hand cases up to n=5
>  >
>  > Unlabeled: 1,1,2,5,8
>  > Labeled:   1,1,2,10,53
> 
> Hmm, I did the same and got
> 
> Unlabeled: 1, 1, 2, 4, 7
> Labeled:   1, 1, 2, 7, 48
> 
> As for the case n=4, I can't see how you get 5/10: The groups I
> see are {1}, the 1 characteristic V_4, the 4 C_3s and the 1 A_4.
> 

I don't know what you mean by "the 1 characteristic V_4," but besides
the (1), C_3 and A_4 I got <(12)(34),(13)(24)> (1 case) and
<(12)(34)> (3 cases)

> Similarly for the case n=5: We get the 5 * (7 - 1) + 1 = 31 groups
> of the already counted isomorphism classes living in one the 5 A_4s
> plus the 10 conjugates of <(1 2 3), (1 2)(4 5)> (isomorphic to S_3)
> plus the 6 conjugates of <(1 2)(3 4), (1 3)(2 5)> (isom. to D_10)
> plus the 1 A_5 itself, giving a total of 31 + 10 + 6 + 1 = 48 groups
> in 4 + 3 = 7 isomorphism classes.

I will look at these.

Christian