# [seqfan] Re: Symmetric groups

Charles Greathouse charles.greathouse at case.edu
Mon Nov 28 18:41:48 CET 2016

This discussion inspired me to submit
https://oeis.org/A277566
the orders of subgroups of S_n.

Charles Greathouse
Case Western Reserve University

On Mon, Nov 28, 2016 at 10:33 AM, Neil Sloane <njasloane at gmail.com> wrote:

> "PS By the way there is a similar
> interesting result about integers n
> such that all groups of order n must
> be abelian (see L. E. Dickson,
> Trans. Amer. Math. Soc. 6 (1905), 198–204).
> I did not check (yet) whether the corresponding
> sequence of integers is in the OEIS: I bet it is!"
>
> Yes, it is A051532
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Email: njasloane at gmail.com
>
>
> On Mon, Nov 28, 2016 at 1:28 AM, jean-paul allouche <
> jean-paul.allouche at imj-prg.fr> wrote:
>
> > The same argument can be used for larger n's.
> > There is only one group of order pq where p
> > and q are two primes such that p < q and
> > q \notequiv 1 \bmod p (this is true more
> > generally for the groups of order n such that
> > n and \phi(n) are coprime, where \phi is the
> > Euler totient).
> >
> > best
> > jp allouche
> >
> > PS By the way there is a similar
> > interesting result about integers n
> > such that all groups of order n must
> > be abelian (see L. E. Dickson,
> > Trans. Amer. Math. Soc. 6 (1905), 198–204).
> > I did not check (yet) whether the corresponding
> > sequence of integers is in the OEIS: I bet it is!
> >
> >
> >
> > Le 28/11/16 à 01:51, israel at math.ubc.ca a écrit :
> >
> > There is only one group of order 15, and it is cyclic.
> >> In order for a member of S_n to have order 15, n must be at least
> >> 8 (so you can have a disjoint 3-cycle and 5-cycle).
> >>
> >> Cheers,
> >> Robert
> >>
> >> On Nov 27 2016, W. Edwin Clark wrote:
> >>
> >> No, S_5 does not have a subgroup of order 15 says GAP. Here's  GAP code
> >>> which gives the orders
> >>> of the subgroups of S_5:
> >>>
> >>> G:=SymmetricGroup(5);;
> >>> C:=ConjugacyClassesSubgroups(G);;
> >>>
> >>>                [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 20, 24, 60, 120 ]
> >>>
> >>> On Sun, Nov 27, 2016 at 4:14 PM, Frank Adams-Watters <
> >>> franktaw at netscape.net> wrote:
> >>>
> >>> If n divides m!, does the symmetric group S_m always have a subgroup of
> >>>> order n?
> >>>>
> >>>> If so, a comment should be added to A002034 that a(n) is the genus of
> >>>> the
> >>>> smallest symmetric group with a subgroup of order n. If not, where is
> >>>> the
> >>>> first exception? (8 in S_4?) Is the sequence so described in the OEIS?
> >>>> If
> >>>> not, it should be added.
> >>>>
> >>>>
> >>>>
> >>>> --
> >>>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>>
> >>>>
> >>> --
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> >>>
> >>>
> >>>
> >> --
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> >>
> >
> >
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>
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