In[]:=
Get[FileNameJoin[{NotebookDirectory[],"GGT.wl"}]]
In[]:=
FiniteGroupData[{"DihedralGroup",4},"NormalSubgroups"]
Out[]=
{{CyclicGroup,1},{CyclicGroup,2},{CyclicGroup,4},{DihedralGroup,2},{DihedralGroup,4}}
In[]:=
FiniteGroupData[{"DihedralGroup",4},"QuotientGroups"]
Out[]=
{Trivial,{CyclicGroup,2},{DihedralGroup,2},{DihedralGroup,4}}
s[n_]:=SymmetricGroup[n];d[n_]:=DihedralGroup[n];c[n_]:=CyclicGroup[n];h=PermutationGroup[{Cycles[{{1,3}}],Cycles[{{2,4}}]}];
IsomorphicGroupQ[h,d[2]](*hisisomorphismtoD2group*)
Out[]=
True
In[]:=
IsomorphicGroupQ[h,c[4]](*D2andC4arenotisomorphic*)
Out[]=
False
In[]:=
(*testiftheyarenormalsubgroupsofD4*)NormalSubgroupQ[h,d[4]]NormalSubgroupQ[c[4],d[4]]
Out[]=
True
Out[]=
True
QuotientGroup[h,d[4]](*ButtheygeneratethesamequotientgroupC2*)QuotientGroup[c[4],d[4]]
Out[]=
PermutationGroup[{Cycles[{}],Cycles[{{1,2}}]}]
Out[]=
PermutationGroup[{Cycles[{}],Cycles[{{1,2}}]}]