and how to prove this make x and y are conjugate by an element of N?

with(GroupTheory):

with(group):

G := AlternatingGroup(3);

IsFinite(G);

GroupOrder(G);

spg := SylowSubgroup(3, G);

IsAbelian(spg);

Elements(spg);

lprint(%);

H := Subgroup(G, spg);

got error, invalid input here,

GroupTheory:-SylowSubgroup(3, module () local labels, minSupp,

maxSupp, suppSize, AtkinsonsAlgorithm, IsSimpleGroupOrder,

doDerivedSeries, doLowerCentralSeries, Intersection2,

RightCosetRepresentatives, LeftCosetRepresentatives, PRA,

`Giant?`, `Even?`, doStab1, doStab, CycleIndexMonomial;

export generator_list, n, supergroup, Sylows, pCores,

ModulePrint, ModuleDeconstruct, Generators, Orbit, Orbits,

IsTransitive, Transitivity, IsPrimitive, GroupOrder,

Elements, IsAbelian, IsElementary, IsSimple, ConjugacyClass,

ConjugacyClasses, CayleyTable, Centre, DerivedSubgroup,

IsPerfect, DerivedSeries, LowerCentralSeries, NilpotencyClass\

, IsNilpotent, doUpperCentralSeries, UpperCentralSeries,

SylowSubgroup, IsSubgroup, IsNormal, Core, NormalClosure,

Normaliser, Conjugator, AreConjugate, Centraliser,

Intersection, `intersect`, LeftCoset, RightCoset,

RightCosets, LeftCosets, Factor, RandomElement, IsAlternating\

, IsSymmetric, PCore, FittingSubgroup, FrattiniSubgroup,

MatrixRepresentation, Stabiliser, CycleIndexPolynomial,

properties; option object; end module)

Error, invalid input: GroupTheory:-Subgroup expects its 1st argument, generators, to be of type {list, set, identical(undefined)}, but received module () local labels, minSupp, maxSupp, suppSize, AtkinsonsAlgorithm, IsSimpleGroupOrder, doDerivedSeries, doLowerCentralSeries, Intersection2, RightCosetRepresentatives, LeftCosetRepresentatives, PRA, `Giant?`, `Even?`, doStab1, doStab, CycleIndexMonomial; export generator_list, n, supergroup, Sylows, pCores, ModulePrint, ModuleDeconstruct, Generators, Orbit, Orbits, IsTransitive, Transitivity, IsPrimitive, GroupOrder, Elements, IsAbelian, IsElementary, IsSimple, ConjugacyClass, ConjugacyClasses, ...

N := Normaliser(G, spg);

H2 := Subgroup(G, G);

how to find the subgroup G which is finite group here?

elist := Elements(H2);

AreConjugate(elist[1], elist[2], N);

**originally x and y are not conjugate,**

how to prove this make x and y are conjugate by an element of N if spg is abelian

where x and y are elements of H2 which is subgroup of G, which is finite group

i guess find subgroup with following command, however, normaliser N can not

make elements of x and y conjugate

H2 := Subgroup(Elements(G), G);

elist := Elements(H2);

AreConjugate(elist[2], elist[3], N); #N*elist[1]*N^(-1) = elist[2]

but it is false,

i use G := AlternatingGroup(5); it is true,

does it mean that this theorem is not for all cases?