A core-free subgroup is a subgroup in which its (normal) core is trivial.
‣ IsCoreFree( G, H ) | ( function ) |
Returns: a boolean
Given a group G and one of its subgroups H, it returns whether H is core-free in G.
gap> G := SymmetricGroup(4);; H := Subgroup(G, [(1,3)(2,4)]);; gap> Core(G,H); Group(()) gap> IsCoreFree(G,H); true gap> H := Subgroup(G, [(1,4)(2,3), (1,3)(2,4)]);; gap> IsCoreFree(G,H); false gap> Core(G,H);# H is a normal subgroup of G, hence it does not have a trivial core Group([ (1,4)(2,3), (1,3)(2,4) ])
‣ CoreFreeConjugacyClassesSubgroups( G ) | ( function ) |
Returns: a list
Returns a list of all conjugacy classes of core-free subgroups of G
gap> G := SymmetricGroup(4);; dh := DihedralGroup(10);; gap> cfccs := CoreFreeConjugacyClassesSubgroups(G);; Size(cfccs); 7 gap> cfccs_dh := CoreFreeConjugacyClassesSubgroups(dh);; Size(cfccs_dh); 2
‣ AllCoreFreeSubgroups( G ) | ( function ) |
Returns: a list
Returns a list of all core-free subgroups of G
gap> G := SymmetricGroup(4);; dh := DihedralGroup(10);; gap> acfs := AllCoreFreeSubgroups(G);; Size(acfs); 24 gap> acfs_dh := AllCoreFreeSubgroups(dh);; Size(acfs_dh); 6
‣ CoreFreeDegrees( G ) | ( function ) |
Returns: a list
Returns a list of all possible degrees of faithful transitive permutation representations of G. The degrees of a faithful transitive permutation representation of G are the index of its core-free subgroups.
gap> G := SymmetricGroup(4);; dh := DihedralGroup(10);; gap> CoreFreeDegrees(G); [ 4, 6, 8, 12, 24 ] gap> CoreFreeDegrees(dh); [ 5, 10 ]
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