[GAP Forum] Finding or determining existence equivalence permutation representations?

[Alexander Hulpke]

Dear Forum, Dear Frederic,

> I had another related question: suppose I have some groups (let us assume of the same size  but not necessarily isomorphic), represented by permutations on sets of the same size.
> Is there any way to determine if the representations are equivalent and if so, to actually see the equivalence?

You want an element in the corresponding symmetric group that conjugates one group to the other:

gap> s8:=SymmetricGroup(8);
Sym( [ 1 .. 8 ] )
gap> RepresentativeAction(s8,g,w[1]);
(2,4,6,3,5,8)
gap> RepresentativeAction(s8,g,w[2]);
fail

> q:=3;
> h:=DirectProduct(CyclicGroup(q),CyclicGroup(q));
> gtemp:=AutomorphismGroup(h);
> Difference(Elements(h),[Elements(h)[1]]);
> g:=Action(gtemp,Difference(Elements(h),[Elements(h)[1]]),OnPoints);
> 
> w:=AllTransitiveGroups(DegreeOperation,q^2-1,Size,(q^2-1)*q*(q-1));
> 
> How could I see which group action in w is equivalent to g?

The transitive groups and primitive groups libraries provide functions
TransitiveIdentification and PrimitiveIdentification
that give the number of the group in the respective library. As they can use the classification of the groups they can avoid the hard isomorphism test and instead check a sufficient number of invariants.

> Would this also be possible for much larger groups?

In most cases RepresentativeAction should work fine for degrees up to the 10000s or even 100000, but I would not be surprised if one could construct particular situations even in degree 1000s where it will take substantial time.

Best,

   Alexander

-- Colorado State University, Department of Mathematics,
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