Dear GAP users, I am using GAP 4.4.7, on a PC with Windows XP.
For my research I need to know the difference between AllGroups(273)[3] and AllGroups(273)[4] and also how I can construct them other than using AllGroups. Sofar I can only construct one and I can't find any group property which is different between the two. My first question is about constructing C91:C3. AllGroups(273) yields. ["C13 x (C7 : C3)", "C7 x (C13 : C3)", "C91 : C3", "C91 : C3", "C273" ] I have been able to construct one instance of C91:C3 through gap> G:=CyclicGroup(IsPermGroup,3); Group([ (1,2,3) ]) gap> N:=CyclicGroup(91); <pc group of size 91 with 2 generators> gap> A:=AutomorphismGroup(N); <group with 2 generators> gap> f:=GroupHomomorphismByImages(G,A,[Elements(G)[2]],[Elements(A)[39]]); [ (1,2,3) ] -> [ Pcgs([ f1, f2 ]) -> [ f1^4*f2^10, f2^9 ] ] gap> NG:=SemiDirectProduct(G,f,N); gap> NG:=SemidirectProduct(G,f,N); <pc group of size 273 with 3 generators> gap> StructureDescription(NG); "C91 : C3" How can I construct the other C91:C3 ? ( I have tried mappings to other elements in A ( C12 x C6 ) but none of them eventually results to a different C91:C3 in the SemidirectProduct. My second question is about the difference is between AllGroups(273)[3] and AllGroups(273)[4], they seem isomorphic to me. I have tested various properties and I can't figure out the difference. About AllGroups(273)[3] and AllGroups(273)[4]. I name the groups G1 and G2... gap> G1:=AllGroups(273)[3]; <pc group of size 273 with 3 generators> gap> G2:=AllGroups(273)[4]; <pc group of size 273 with 3 generators> If the subgroups of G1, G2 are the same and the orders of the elements... then where is the difference? I can't find it in GAP. The following commands illustrate some significant similarities between the groups. gap> StructureDescription(G1); "C91 : C3" gap> StructureDescription(G2); "C91 : C3" gap> List(ConjugacyClassesSubgroups(G1),Representative); [ Group([ ]), C3, Group([ f2 ]), Group([ f3 ]), Group([ f1, f2 ]), Group([ f3, f1 ]), C91, Group([ f3, f2, f1 ]) ] gap> List(List(ConjugacyClassesSubgroups(G1),Representative),StructureDescript$ gap> $cription); [ "1", "C3", "C7", "C13", "C7 : C3", "C13 : C3", "C91", "C91 : C3" ] gap> List(List(ConjugacyClassesSubgroups(G2),Representative),StructureDescript$ gap> $cription); [ "1", "C3", "C7", "C13", "C7 : C3", "C13 : C3", "C91", "C91 : C3" ] gap> $cription); [ "1", "C3", "C7", "C13", "C7 : C3", "C13 : C3", "C91", "C91 : C3" ] gap> Sum(List(Elements(G1),Order)); 7297 gap> Sum(List(Elements(G2),Order)); 7297 Thanks on beforehand for any hints on this particular issue. -- met vriendelijke groet / kind regards, nilo de roock _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
