# [seqfan] Re: Symmetric groups

jean-paul allouche jean-paul.allouche at imj-prg.fr
Mon Nov 28 21:06:15 CET 2016

Thank you Neil! (I checked myself inbetween)
with my best wishes
jean-paul

Le 28/11/16 à 16:33, Neil Sloane a écrit :
> "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
>>>>>
>>>>>
>>>>> --
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>>>>>
>>>>>
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>>>>
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>>
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