[seqfan] Re: Symmetric groups
Neil Sloane
njasloane at gmail.com
Mon Nov 28 19:03:43 CET 2016
I added a link to A277566 to point to this useful web page:
Götz Pfeiffer, <a href="http://schmidt.nuigalway.ie/subgroups">Numbers of
subgroups of various families of groups</a>
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.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com
On Mon, Nov 28, 2016 at 12:41 PM, Charles Greathouse <
charles.greathouse at case.edu> wrote:
> 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.
> > Phone: 732 828 6098; home page: http://NeilSloane.com
> > 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);;
> > >>> OrdersSubgroupsS_5:=Set(ListX(C,t->Size(Representative(t))));
> > >>>
> > >>> [ 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.
> > >>>>
> > >>>> Franklin T. Adams-Watters
> > >>>>
> > >>>>
> > >>>> --
> > >>>> Seqfan Mailing list - http://list.seqfan.eu/
> > >>>>
> > >>>>
> > >>> --
> > >>> Seqfan Mailing list - http://list.seqfan.eu/
> > >>>
> > >>>
> > >>>
> > >> --
> > >> Seqfan Mailing list - http://list.seqfan.eu/
> > >>
> > >
> > >
> > > --
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> > >
> >
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> >
>
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