[GAP Forum] Minimal faithful permutation representation degree

[Holt, Derek]

Dear Forum

In case anyone is interested, here is an example in which the inequality is
strict - that is the minimal degree of a faithful permutation of G x H is less
than the sum of the minimal degrees of G and H. Fortunately, GAP is getting the
answers right in this example!

gap> x:=(1, 2, 3)(4, 6, 5)(7, 8, 9)(10, 11, 12);;
gap> y:=(1, 5, 9, 11)(2, 12, 7, 4);;
gap> z:= (2, 4, 7, 12)(3, 10, 8, 6);;
gap> H := Group([x,y,z]);;
gap> G:=CyclicGroup(2);;
gap> D:=DirectProduct(G,H);;
gap> MinimalFaithfulPermutationDegree(G);
2
gap> MinimalFaithfulPermutationDegree(H);        
12
gap> MinimalFaithfulPermutationDegree(D);
12

Regards,
Derek Holt

On Sat, Aug 31, 2019 at 08:41:57PM +0000, Hulpke,Alexander wrote:
> Dear Forum,
> 
> > 
> > Let G be any finite group, let $\mu G$ be the minimal faithful permutation representation degree of G, all research papers  I got trying to investigate whether the inequality $\mu G\times \mu H \leq \mu G +\mu H$ is strict or equality, where H is any other group, I did the following and wish somebody explain;
> > 
> > 
> > gap> a:=AlternatingGroup(5);:
> > gap> IsPerfectGroup(a);
> > true
> > gap> s:=OneSmallGroup(46,IsSolvable,true);
> > <pc group of size 46 with 2 generators>
> > gap> D:=DirectProduct(s,a);
> > <group of size 2760 with 4 generators>
> > gap> mua:=MinimalFaithfulPermutationDegree(a);
> > 5
> > gap> mus:=MinimalFaithfulPermutationDegree(s);
> > 23
> > gap> muD:=MinimalFaithfulPermutationDegree(D);
> > 33
> > 
> 
> Thank you for reporting. This is a bug that will be corrected in a future release.
> 
> Regards,
> 
>   Alexander Hulpke
> 
> 
> 
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