# [seqfan] Re: Symmetric group S_n as product of at most A225788(n) cyclic subgroups.

L. Edson Jeffery lejeffery2 at gmail.com
Fri May 24 02:22:35 CEST 2013

```>On Thu, May 23, 2013 at 3:58 PM, Charles Greathouse <
charles.greathouse at case.edu> wrote:
>
>I think the sequence can be defined unambiguously: Smallest k such that the
>symmetric group S_n is a product of at most k cyclic subgroups. I would
>love to see this sequence in the OEIS. Do we have any GAP experts who can
>code this up?
>
>Charles Greathouse
>Analyst/Programmer
>Case Western Reserve University

Charles,

In retrospect, the definition "a(n) = least k such that the symmetric group
S_n is a product of k cyclic subgroups" might be a better one (if the
sequence can be determined) because of the following. For this idea, Abért
gave the lower bound of (1 + o(1))*(n*log(n))^(1/2) cyclic subgroups.  But
for the upper bound (A225788) he pointed out the "obvious decomposition"

S_n = [(1,2,...,n)][(1,2,...,n-1)]...[(1,2)]

of S_n as a product of n-1 cyclic subgroups (replace the brackets [] with
angled ones). Although I don't claim to understand all of this to any great
depth, it appears that your idea for the sequence would reduce to just a(n)
= n-1 which unfortunately does not seem terribly interesting. Can you use
the above lower bound to produce another sequence like A255788?

Ed Jeffery
```