# [seqfan] Re: Symmetric groups

Neil Sloane njasloane at gmail.com
Mon Nov 28 16:33:57 CET 2016

"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);;
>>>
>>>                [ 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
>>>>
>>>>
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>>>>
>>>>
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>>>
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>
>
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