Please note, that throughout this package it is assumed that the group G is finite.
‣ BasicGroup( KG ) | ( function ) |
Let G be a group and F a field. Then the group algebra FG consists of the set of formal linear combinations of the form
\sum_{g \in G} \alpha_g g, \quad \alpha_g \in F
where all but finitely many of the α_g are zero [Lee73][Bov98].
For the group ring KG the function BasicGroup returns the basic group of KG as a subgroup of the group of units of KG.
gap> G:=CyclicGroup(IsFpGroup,4); <fp group of size 4 on the generators [ a ]> gap> Elements(G); [ <identity ...>, a, a^2, a^3 ] gap> KG:=GroupRing(GF(2),G); <algebra-with-one over GF(2), with 1 generators> gap> BasicGroup(KG); <group with 4 generators> gap> Elements(last); [ (Z(2)^0)*<identity ...>, (Z(2)^0)*a, (Z(2)^0)*a^2, (Z(2)^0)*a^3 ]
‣ IsLienEngel( KG ) | ( function ) |
On any associative ring R the Lie bracket is defined as [x,y] = xy-yx.. For any positive integer n, a subset Λ of R is said to be Lie n-Engel if for any x,y ∈ Λ, [x,y,...,y]=0 (y is taken n times) [Lee00][Seh78] .
For the group ring KG the function IsLienEngel returns true if KG is Lie n-Engel and false otherwise.
gap> G:=CyclicGroup(4); <pc group of size 4 with 2 generators> gap> KG:=GroupRing(GF(2),G); <algebra-with-one over GF(2), with 2 generators> gap> IsLienEngel(KG); false gap> G:=DihedralGroup(16); <pc group of size 16 with 4 generators> gap> KG:=GroupRing(GF(2),G); <algebra-with-one over GF(2), with 4 generators> gap> IsLienEngel(KG); true
‣ GetRandomUnit( KG ) | ( function ) |
For the group ring KG the function GetRandomUnit returns an unit (i.e. an invertible element) by a random way.
gap> G:=CyclicGroup(4); <pc group of size 4 with 2 generators> gap> KG:=GroupRing(GF(7),G); <algebra-with-one over GF(7), with 2 generators> gap> u:=GetRandomUnit(KG);; gap> Augmentation(u); Z(7)^4 gap> u*u^-1; (Z(7)^0)*<identity> of ...
‣ GetRandomNormalizedUnit( KG ) | ( function ) |
Let U(KG) be the multiplicative group of the group algebra KG of the p-group G over a field K of characteristic p. The subgroup
V(KG)=\{ \sum_{g \in G} \alpha_g g \in U(KG) \mid \sum_{g \in G} \alpha_g = 1 \}
is called the group of normalized units of the group algebra KG [BS89].
For the group ring KG the function GetRandomNormalizedUnit returns a normalized unit (i.e. an invertible element with augmentation 1, u ∈ V(KG)) by a random way.
gap> G:=DihedralGroup(IsFpGroup,16);; gap> KG:=GroupRing(GF(2),G);; gap> u:=GetRandomNormalizedUnit(KG);; gap> Augmentation(u); Z(2)^0 gap> u*u^-1; (Z(2)^0)*<identity ...>
‣ GetRandomNormalizedUnitaryUnit( KG[, eta] ) | ( function ) |
Let p be a prime, K the field of characteristic p, G a finite abelian p-group with an arbitrary antiautomorphism η of order 2 and G_η={g ∈ G ∣ η(g)=g}. Extending the antiautomorphism η to the group algebra KG we obtain the involution
x = \sum_{g \in G} \alpha_g g \mapsto x^{*}= \sum_{g \in G} \alpha_g \eta(g)
of KG which is called as η-involution. If η(g)=g^-1 for all g ∈ G then this involution is called canonical. Then, if V(KG) is the group of normalized units of KG, the subgroup of unitary units is defined as [BS06]
V_*(KG)=\{x \in V(FG) \mid x^* = x^{-1} \}.
For the group algebra KG the function GetRandomNormalizedUnitaryUnit returns a normalized unitary unit (i.e. such an invertible element with augmentation 1, that u⋅ u^*=1) by a random way. Also, there exists a two-parametrical version of this method, where the second parameter η is an arbitrary antiautomorphism of G. In this case, the function GetRandomNormalizedUnitaryUnit returns a normalized unitary unit (i.e. such an invertible element with augmentation 1, that u⋅ u^*=1) by a random way for the group algebra KG and an arbitrary antiautomorphism eta.
gap> G:=CyclicGroup(4);; gap> KG:=GroupRing(GF(2),G);; gap> u:=GetRandomNormalizedUnitaryUnit(KG);; gap> u*Involution(u); (Z(2)^0)*<identity> of ... gap> Augmentation(u); Z(2)^0 gap> Aut:=AutomorphismGroup(G);; gap> KG:=GroupRing(GF(2),G);; gap> Aut2:=Filtered(Aut,i->Order(i)=2);; gap> Aut2; [ [ f1*f3 ] -> [ f1*f2*f4 ], [ f1*f3 ] -> [ f1*f3*f4 ], [ f1*f3 ] -> [ f1*f2 ] ] gap> eta:=Aut2[2]; [ f1*f3 ] -> [ f1*f3*f4 ] gap> u:=GetRandomNormalizedUnitaryUnit(KG,eta);; gap> u*RAMEGA_InvolutionKG(u, eta, KG); (Z(2)^0)*<identity> of ...
‣ GetRandomCentralNormalizedUnit( KG ) | ( function ) |
For the group ring KG the function GetRandomCentralNormalizedUnit returns a central normalized unit (i.e. such an invertible element with augmentation 1, that u⋅ x=x ⋅ u, ∀ x ∈ KG) by a random way.
gap> G:=CyclicGroup(IsFpGroup,4);; gap> KG:=GroupRing(GF(2),G);; gap> u:=GetRandomCentralNormalizedUnit(KG);; gap> Augmentation(u); Z(2)^0 gap> bool:=true; true gap> for x in Elements(KG) do > if x*u<>u*x then bool:=false; break; fi; > od; gap> bool; true
‣ GetRandomElementFromAugmentationIdeal( KG ) | ( function ) |
If KG is a modular group ring then its augmentation ideal A is generated by all elements of the form g-1 where g ∈ G ∖ {1 }. The augmentation ideal consists of all elements with augmentation 0, i.e.
A(KG)=\{ \sum_{g \in G} \alpha_g g \in KG \mid \sum_{g \in G} \alpha_g = 0 \}.
For the modular group ring KG the function GetRandomElementFromAugmentationIdeal returns an element from the augmentation ideal of KG by a random way.
gap> G:=QuaternionGroup(16); <pc group of size 16 with 4 generators> gap> KG:=GroupRing(GF(2),G); <algebra-with-one over GF(2), with 4 generators> gap> u:=GetRandomElementFromAugmentationIdeal(KG);; gap> Augmentation(u); 0*Z(2)
‣ RandomLienEngelLength( KG, num ) | ( function ) |
Let KG be a Lie n-Engel group ring and let [x,y,y,...,y]=0 for all x, y ∈ KG. Then the number of y's in the last equation is called the Lie n-Engel length [BK77][Kur96][Kur98].
For the Lie n-Engel group ring KG and the maximal number of iterations num the function RandomLienEngelLength returns the Lie n-Engel length of KG by a random way.
gap> G:=DihedralGroup(16);; gap> KG:=GroupRing(GF(2),G);; gap> RandomLienEngelLength(KG,100); 4
‣ RandomExponent( KG, num ) | ( function ) |
For the group ring KG (where G is a finite p-group and the characteristic of K is p) and the maximal number of iterations num the function RandomExponent returns the exponent of V(KG) [BL93] [BS18] [Sha91] by a random way.
gap> G:=DihedralGroup(16);; gap> KG:=GroupRing(GF(2),G);; gap> RandomExponent(KG,100); 8
‣ RandomExponentOfNormalizedUnitsCenter( KG, num ) | ( function ) |
For the group ring KG (where G is a finite p-group and the characteristic of K is p) and the maximal number of iterations num the function RandomExponentOfNormalizedUnitsCenter returns the exponent of the center of the group of normalized units of KG by a random way.
gap> G:=DihedralGroup(16);; gap> KG:=GroupRing(GF(2),G);; gap> RandomExponentOfNormalizedUnitsCenter(KG,100); 4
‣ RandomNilpotencyClass( KG, num ) | ( function ) |
Let G be a group and let g_1,...,g_n ∈ G. By the symbol (g_1,...,g_n) we denote the commutator of the elements g_1,...,g_n which is defined inductively as
(g_1,\ldots,g_n)=((g_1,\ldots,g_{n-1}),g_n), \quad (g_1,g_2)=g_1^{-1}g_2^{-1}g_1 g_2.
As usual, for the subsets X, Y of G we will denote by (X,Y) the subgroup generated by all commutators (x,y) with x ∈ X, y ∈ Y.
This allows us to define the lower central series of a nonempty subset H of G: let γ_n+1(H)=(γ_n(H),H) with γ_1(H)=H. We say that H is nilpotent if γ_n(H)=1 for some n. For a nilpotent subset H ⊆ G the number
cl(H) = \min\{n \in \mathbb{N}_0: \gamma_{n+1}(H)=1\}
is called the nilpotency class of H [BJ11][Sha93][Sha91][MS90] [Sha90b][Sha90a].
For the Lie nilpotent group ring KG and the maximal number of iterations num the function RandomNilpotencyClass returns the nilpotency class of the group of normalized units of KG by a random way.
gap> G:=DihedralGroup(16);; gap> KG:=GroupRing(GF(2),G);; gap> RandomNilpotencyClass(KG,100); 4
‣ RandomDerivedLength( KG, n ) | ( function ) |
KG is called Lie solvable, if some of the terms of the Lie derived series δ^[n](KG)=[δ^[n-1](KG),δ^[n-1](KG)] with δ^[0](KG)=KG are equal to zero.
Denote by dl_L(KG) the minimal element of the set {m∈ N | δ^[m](KG)=0}, which is said to be the Lie derived length of KG [Bal12] [BT08][Spi08][CS10].
For the modular group ring KG and a positive integer n the function RandomDerivedLength returns the Lie derived length by a random way.
gap> D:=DihedralGroup(IsFpGroup,8);; gap> KG:=GroupRing(GF(2),D);; gap> RandomDerivedLength(KG,100); 2
‣ RandomCommutatorSubgroup( KG, n ) | ( function ) |
For the group ring KG and a positive integer n the function RandomCommutatorSubgroup returns the commutator subgroup of the unit group of KG using n iterations by a random way. There is also an overloaded version of this method for finding the commutator subgroup of an arbitrary group G using n iterations by a random way.
gap> G:=DihedralGroup(IsFpGroup,8);; gap> KG:=GroupRing(GF(2),G);; gap> SG:=RandomCommutatorSubgroup(KG,100); gap> StructureDescription(SG); "C2 x C2 x C2"
‣ RandomCommutatorSubgroupOfNormalizedUnits( KG, n ) | ( function ) |
For the modular group ring KG and a positive integer n the function RandomCommutatorSubgroupOfNormalizedUnits returns the commutator subgroup of normalized units by a random way.
gap> G:=DihedralGroup(8);; gap> KG:=GroupRing(GF(2),G);; gap> SG:=RandomCommutatorSubgroupOfNormalizedUnits(KG,100);; gap> u:=Random(Elements(SG));; gap> Augmentation(u); Z(2)^0
‣ RandomNormalizedUnitGroup( KG, n ) | ( function ) |
For a group ring KG, where G is a finite p-group and K has characteristic p and a positive integer n, where n is the number of iterations, the function RandomNormalizedUnitGroup returns the normalized unit group of KG by a random way.
gap> G:=DihedralGroup(8);; gap> KG:=GroupRing(GF(2),G);; gap> SG:=RandomNormalizedUnitGroup(KG); <group with 4 generators> gap> Size(SG); 128 gap> u:=Random(Elements(SG));; gap> Augmentation(u); Z(2)^0
‣ RandomUnitarySubgroup( KG, n ) | ( function ) |
For a modular group ring KG and a positive integer n, where n is the number of iterations, the function RandomUnitarySubgroup returns the subgroup generated by the normalized unitary units (see GetRandomNormalizedUnitaryUnit (2.2-3)) by a random way.
gap> G:=DihedralGroup(8);; gap> KG:=GroupRing(GF(2),G);; gap> SG:=RandomUnitarySubgroup(KG,100);; gap> u:=Random(Elements(SG));; gap> Augmentation(u); Z(2)^0 gap> u*u^-1; (Z(2)^0)*<identity> of ...
‣ RandomCommutatorSeries( KG, n ) | ( function ) |
For a modular group ring KG, where G is a p-group, and a positive integer n, where n is the number of iterations, the function RandomCommutatorSeries returns the commutator series of the group of normalized units in KG by a random way.
gap> G:=DihedralGroup(8);; gap> KG:=GroupRing(GF(2),G);; gap> CS:=RandomCommutatorSeries(KG,100); [ <group of size 128 with 4 generators>, <group of size 8 with 8 generators>, <group of size 1 with 1 generators> ]
‣ RandomLowerCentralSeries( KG, n ) | ( function ) |
The lower central series of a group G is the descending series of groups
G=G_1 \trianglerighteq G_2 \trianglerighteq \ldots \trianglerighteq G_n \trianglerighteq \ldots,
where each G_n+1=[G_n,G], the subgroup generated by all commutators [x,y] with x in G_n and y in G. Thus, G_2=[G,G]=G^(1), the derived subgroup of G; G_3=[[G,G],G], etc. The lower central series is often denoted y_n(G).
For a group ring KG, for which V(KG) is nilpotent, and a positive integer n, where n is the number of iterations, the function RandomLowerCentralSeries returns the lower central series of the group of normalized units of KG by a random way.
gap> G:=DihedralGroup(8);; gap> KG:=GroupRing(GF(2),G);; gap> CS:=RandomLowerCentralSeries(KG,100); [ <group of size 128 with 4 generators>, <group of size 8 with 8 generators>, <group of size 1 with 1 generators> ]
‣ RandomUnitaryOrder( KG, n[, sigma] ) | ( function ) |
For the group ring KG, where G is a finite p-group and K has characteristic p, and a positive integer n, where n is the number of iterations, the function RandomUnitaryOrder returns the order of the unitary subgroup of KG [CG09] by a random way. Also, there exists a three-parametrical version of this method, where the third parameter sigma is an arbitrary anti-automorphism of G.
gap> G:=DihedralGroup(8);; gap> KG:=GroupRing(GF(2),G);; gap> ord:=RandomUnitaryOrder(KG,100); 64
‣ RandomOmega( G, m, n ) | ( function ) |
For the p group G and a positive integers m and n (number of iterations) the function RandomOmega returns the subgroup of G, that is generated by the elements g for which g^p^m=1, by random search.
gap> G:=CyclicGroup(8);; gap> KG:=GroupRing(GF(2),G);; gap> RO:=RandomOmega(KG,2,50);; gap> u:=Random(Elements(RO));; gap> (u^2)^2; (Z(2)^0)*<identity> of ... gap> ## Let h be a group. gap> h:=SmallGroup(16,10);; gap> RO:=RandomOmega(h,2,50);; gap> u:=Random(Elements(RO));; gap> (u^2)^2; <identity> of ...
‣ RandomAgemo( G, m, n ) | ( function ) |
For the p-group G and a positive integers m and n (number of iterations) the function RandomOmega returns the subgroup of G, that is generated by the elements g^p^m by random search.
gap> G:=CyclicGroup(8);; gap> KG:=GroupRing(GF(2),G);; gap> RA:=RandomAgemo(KG,2,50);; gap> Elements(RA); [ (Z(2)^0)*<identity> of ..., (Z(2)^0)*f3 ] gap> u:=Random(Elements(RA));; gap> (u^2)^2; (Z(2)^0)*<identity> of ... gap> ## Let h be a group. gap> h:=SmallGroup(16,10);; gap> RA:=RandomAgemo(h,2,50);; gap> Elements(RA); [ <identity> of ... ]
‣ RandomDihedralDepth( G, n ) | ( function ) |
For a non abelian finite 2-group G and a positive integer n, where n is the number of iterations, the function RandomDihedralDepth returns the dihedral depth of G, which is defined to be the maximal number d such that G contains a subgroup isomorphic to the dihedral group of order 2^(d+1).
gap> G:=SmallGroup([64,6]); <pc group of size 64 with 6 generators> gap> StructureDescription(G); "(C8 x C4) : C2" gap> RandomDihedralDepth(G,1000); 2
‣ RandomQuaternionDepth( G, n ) | ( function ) |
For a non abelian finite 2-group G and a positive integer n, where n is the number of iterations, the function RandomQuaternionDepth returns the quaternion depth of G, which is defined to be the maximal number d such that G contains a subgroup isomorphic to the generalized quaternion group of order 2^(d+1).
gap> G:=SmallGroup([64,10]); <pc group of size 64 with 6 generators> gap> StructureDescription(G); "(C8 : C4) : C2" gap> RandomQuaternionDepth(G,1000); 2
‣ GetRandomSubgroupOfNormalizedUnitGroup( KG, n ) | ( function ) |
For a group ring KG and a positive integer n, the function GetRandomSubgroupOfNormalizedUnitGroup returns a subgroup of the normalized group of units generated by n random normalized units.
gap> KG:=GroupRing(GF(2),DihedralGroup(8));; gap> G:=GetRandomSubgroupOfNormalizedUnitGroup(KG,2); <group with 2 generators> gap> StructureDescription(G); "(C4 x C2) : C2"
‣ RandomConjugacyClass( KG, n ) | ( function ) |
For a group ring KG and a positive integer n (number of iterations), the function RandomConjugacyClass returns the random conjugacy class of a random element of the group of normalized units.
gap> KG:=GroupRing(GF(2),DihedralGroup(8));; gap> cc:=RandomConjugacyClass(KG,4);; gap> List(cc,i->Order(i)); [ 4, 4, 4 ]
‣ RandomConjugacyClasses( KG, n ) | ( function ) |
For a modular group algebra KG and a positive integer n (number of iterations), the function RandomConjugacyClasses returns the conjugacy classes of the group of normalized units.
gap> KG:=GroupRing(GF(2),SmallGroup(8,3));; gap> cc:=RandomConjugacyClasses(KG,100);; gap> Size(Union(cc)); 128 gap> Collected(List(cc,x->Number(x))); [ [ 1, 16 ], [ 4, 28 ] ] gap> vkg:=RandomNormalizedUnitGroup(KG);; gap> Size(vkg); 128
‣ RandomIsCentralElement( KG, u, n ) | ( function ) |
For a group ring KG, element u ∈ KG, and a positive integer n (number of iterations), the function RandomIsCentralElement returns true if u is a central element by random way.
gap> KG:=GroupRing(GF(2),CyclicGroup(16));; gap> u:=Random(Elements(KG));; gap> RandomIsCentralElement(KG,u,100); true
‣ RandomIsNormal( KG, N, n ) | ( function ) |
For a group ring KG, subgroup N, and a positive integer n (number of iterations), the function RandomIsNormal returns true if N is normal in the normalized group of units by random way.
gap> KG:=GroupRing(GF(2),CyclicGroup(4));; gap> u:=Elements(KG)[16]; (Z(2)^0)*f1*f2 gap> G:=Group(u);; gap> StructureDescription(G); "C4" gap> RandomIsNormal(KG,G,100); true
‣ RandomCenterOfCommutatorSubgroup( KG, n ) | ( function ) |
For a group ring KG, and a positive integer n (number of iterations), the function RandomCenterOfCommutatorSubgroup returns the center of the commutator subgroup of the normalized unit group using random method.
gap> RANDOM_SEED(1); gap> KG:=GroupRing(GF(2),DihedralGroup(8));; gap> cc:=RandomCenterOfCommutatorSubgroup(KG,100); <group with 8 generators> gap> StructureDescription(cc); "C2 x C2 x C2"
‣ RandomConjugacyClassByElement( KG, u, n ) | ( function ) |
For a group ring KG, normalized unit u ∈ KG, and a positive integer n (number of iterations), the function RandomConjugacyClassByElement returns the random conjugacy class of the normalized unit u.
gap> KG:=GroupRing(GF(2),DihedralGroup(8));; gap> u:=Random(Elements(KG));; gap> rcc: = RandomConjugacyClassByElement(KG,u,100); gap> rcc[1]*u*rcc[1]^-1; true
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