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 > > > > _______________________________________________ > Forum mailing list > Forum@gap-system.org > https://mail.gap-system.org/mailman/listinfo/forum _______________________________________________ Forum mailing list Forum@gap-system.org https://mail.gap-system.org/mailman/listinfo/forum
