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Add NormalSubgroups methods for Sym and Alt groups
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‎lib/gpprmsya.gi‎

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@@ -2447,6 +2447,30 @@ InstallMethod( RadicalGroup, "symmetric", true,
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InstallMethod( RadicalGroup, "alternating", true,
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[ IsNaturalAlternatingGroup and IsFinite], 0,RadicalSymmAlt);
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InstallMethod(NormalSubgroups,
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"for a symmetric group",
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[IsSymmetricGroup],
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RankFilter(IsPermGroup),
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function(S)
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if SymmetricDegree(S) <= 4 then
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# S is soluble, so this includes the trivial group (and Klein 4)
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return DerivedSeriesOfGroup(S);
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fi;
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# DerivedSubgroup is the alternating group
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return [S, DerivedSubgroup(S), TrivialSubgroup(S)];
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end);
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InstallMethod(NormalSubgroups,
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"for an alternating group",
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[IsAlternatingGroup],
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RankFilter(IsPermGroup),
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function(A)
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if AlternatingDegree(A) <= 4 then
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# S is soluble, so this includes the trivial group (and Klein 4)
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return DerivedSeriesOfGroup(A);
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fi;
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return [A, TrivialSubgroup(A)];
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end);
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#############################################################################
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##

‎tst/testinstall/opers/NormalSubgroups.tst‎

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gap> START_TEST("NormalSubgroups.tst");
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# Natural symmetric groups
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gap> NormalSubgroups(SymmetricGroup(0)) = [Group(())];
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true
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gap> NormalSubgroups(SymmetricGroup(1)) = [Group(())];
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true
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gap> Set(NormalSubgroups(SymmetricGroup(2))) =
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> Set([Group(()), SymmetricGroup(2)]);
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true
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gap> Set(NormalSubgroups(SymmetricGroup(3))) =
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> Set([Group(()), AlternatingGroup(3), SymmetricGroup(3)]);
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true
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gap> Set(NormalSubgroups(SymmetricGroup(4))) =
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> Set([Group(()), Group((1,2)(3,4), (1,3)(2,4)),
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> AlternatingGroup(4), SymmetricGroup(4)]);
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true
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gap> Set(NormalSubgroups(SymmetricGroup(5))) =
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> Set([Group(()), AlternatingGroup(5), SymmetricGroup(5)]);
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true
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gap> Set(NormalSubgroups(SymmetricGroup(6))) =
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> Set([Group(()), AlternatingGroup(6), SymmetricGroup(6)]);
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true
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gap> Set(NormalSubgroups(SymmetricGroup(100))) =
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> Set([Group(()), AlternatingGroup(100), SymmetricGroup(100)]);
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true
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gap> Set(NormalSubgroups(SymmetricGroup([3,7,4,11,70]))) =
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> Set([Group(()), AlternatingGroup([3, 4, 7, 11, 70]),
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> SymmetricGroup([3, 4, 7, 11, 70])]);
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true
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gap> Set(NormalSubgroups(SymmetricGroup([7,4,13,21]))) =
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> Set([Group(()), Group((4,7)(13,21), (4,13)(7,21)),
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> AlternatingGroup([4, 7, 13, 21]), SymmetricGroup([4, 7, 13, 21])]);
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true
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gap> NormalSubgroups(SymmetricGroup([42])) = [Group(())];
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true
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gap> NormalSubgroups(SymmetricGroup([])) = [Group(())];
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true
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gap> G := Group((7,3,5,11), (11,3), (5,3));;
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gap> IsNaturalSymmetricGroup(G);
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true
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gap> Set(NormalSubgroups(G)) =
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> Set([Group(()), Group((3,5)(7,11), (3,7)(5,11)),
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> AlternatingGroup([3, 5, 7, 11]), SymmetricGroup([3, 5, 7, 11])]);
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true
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# Non-natural symmetric groups
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gap> S4 := Group((1,2,3,4), (1,2));;
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gap> hom := ActionHomomorphism(g, Arrangements([1..4], 4), OnTuples);;
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gap> G := Image(hom);;
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gap> IsSymmetricGroup(G);
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true
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gap> SymmetricDegree(G);
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4
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gap> MovedPoints(G) = [1..24];
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true
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gap> Set(NormalSubgroups(G)) = Set([G,
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> Group([(1,13,9)(2,14,10)(3,15,7)(4,16,8)
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> (5,17,12)(6,18,11)(19,23,22)(20,24,21),
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> (1,21,16)(2,22,15)(3,19,18)(4,20,17)
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> (5,24,13)(6,23,14)(7,11,10)(8,12,9)]),
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> Group([(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)
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> (7,18)(8,17)(9,16)(10,15)(11,14)(12,13),
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> (1,8)(2,7)(3,11)(4,12)(5,9)(6,10)
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> (13,21)(14,22)(15,19)(16,20)(17,24)(18,23)]),
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> Group(())]);
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true
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# Natural alternating groups
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gap> NormalSubgroups(AlternatingGroup(0)) = [Group(())];
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true
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gap> NormalSubgroups(AlternatingGroup(1)) = [Group(())];
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true
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gap> Set(NormalSubgroups(AlternatingGroup(2))) =
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> Set([Group(()), AlternatingGroup(2)]);
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true
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gap> Set(NormalSubgroups(AlternatingGroup(3))) =
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> Set([Group(()), AlternatingGroup(3)]);
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true
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gap> Set(NormalSubgroups(AlternatingGroup(4))) =
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> Set([Group(()), Group((1,2)(3,4), (1,3)(2,4)),
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> AlternatingGroup(4)]);
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true
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gap> Set(NormalSubgroups(AlternatingGroup(5))) =
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> Set([Group(()), AlternatingGroup(5)]);
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true
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gap> Set(NormalSubgroups(AlternatingGroup(6))) =
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> Set([Group(()), AlternatingGroup(6)]);
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true
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gap> Set(NormalSubgroups(AlternatingGroup(100))) =
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> Set([Group(()), AlternatingGroup(100)]);
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true
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gap> Set(NormalSubgroups(AlternatingGroup([3,7,4,11,70]))) =
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> Set([Group(()), AlternatingGroup([3, 4, 7, 11, 70])]);
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true
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gap> Set(NormalSubgroups(AlternatingGroup([7,4,13,21]))) =
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> Set([Group(()), Group((4,7)(13,21), (4,13)(7,21)),
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> AlternatingGroup([4, 7, 13, 21])]);
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true
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gap> NormalSubgroups(AlternatingGroup([42])) = [Group(())];
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true
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gap> NormalSubgroups(AlternatingGroup([])) = [Group(())];
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true
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gap> G := Group((7,3,5), (11,3,7));;
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gap> IsNaturalAlternatingGroup(G);
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true
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gap> Set(NormalSubgroups(G)) =
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> Set([Group(()), Group((3,5)(7,11), (3,7)(5,11)),
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> AlternatingGroup([3, 5, 7, 11])]);
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true
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#
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gap> STOP_TEST( "NormalSubgroups.tst", 1);

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