[seqfan] Symmetric group S_n as product of at most A225788(n) cyclic subgroups.
L. Edson Jeffery
lejeffery2 at gmail.com
Thu May 23 21:56:43 CEST 2013
According to Miklós Abért (see A225788), the symmetric group S_n is a
product of at most 72*n^(1/2)*(log(n))^(3/2) cyclic subgroups. I took the
floor() of this expression and recently submitted the sequence as A225788.
Neil Sloane asked if the sequence of true values is in OEIS, that is, the
sequence with definition "S_n is the product of (exactly) a(n) cyclic
subgroups."
I tried to work on this but failed. It is not even clear to me that the
sequence can be determined unambiguously because if S_n = G_1 X G_2 X ... X
G_k, for some k > 1, then it seems possible that also S_n = H_1 x ... x
H_j, for some j != k, where the G's and H's are cyclic subgroups, and
(where it should be that) j and k depend on the factorization of n. If not
in OEIS and someone would like to figure this out and submit it, that would
be greatly appreciated. Otherwise, if someone finds the sequence in the
database, then either way please let me know the A-number so I can point
A225788 to it. Thanks.
Ed Jeffery
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