[seqfan] Re: Symmetric groups

[israel at math.ubc.ca]

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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>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
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