[GAP Forum] Stabilizer in S_n of an equipartition

[Alexander Hulpke]

Dear Forum, Dear Joe,

On Feb 11, 2009, at 2/11/09 12:03, Joe Bohanon wrote:

> How would I get the stabilizing subgroup in S_n of an equipartition  
> of n (ie
> one where all parts have the same size)?  For instance, in S_15, the
> stabilizer of [1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15] would be  
> the
> wreath product of S_3 with S_5.
>
> The only thing I can think of right now is to take the standard wreath
> product and figure out what the conjugating element would be, but  
> that's too
> much work once n is big.

That's what I would do (it is in fact used inside the routine that  
calculates the normalizer within the symmetric group:  
`NormalizerParentSA' in lib/gpprmsya.gi).

Finding such an conjugating element is rather cheap in the full  
symmetric group:
E.g. suppose that p is the equipartition:
[ [ 1, 5, 24, 27, 29 ], [ 2, 3, 4, 18, 30 ], [ 6, 13, 20, 26, 28 ],
   [ 7, 9, 14, 15, 21 ], [ 8, 12, 17, 19, 23 ], [ 10, 11, 16, 22, 25 ] ]
Form the wreath product:
w 
:=WreathProduct(SymmetricGroup(Length(p[1])),SymmetricGroup(Length(p)));

``Standard'' partition:
sp:=Blocks(w,MovedPoints(w));
[ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8, 9, 10 ], [ 11, 12, 13, 14, 15 ],
   [ 16, 17, 18, 19, 20 ], [ 21, 22, 23, 24, 25 ], [ 26, 27, 28, 29,  
30 ] ]

Mapping element:
map:=MappingPermListList(Concatenation(sp),Concatenation(p));
(2,5,29,22,12,13,20,21,8,4,27,11,6)(3,24,19,15,28,16,7) 
(9,18,14,26,10,30,25,
23,17)

Stabilizing group:
stb:=w^map;

Best,

    Alexander



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