‣ IsMinimalNonAbelianGroup( G ) | ( property ) |
Returns: a boolean
The argument is a group \(G\). The output is true if \(G\) is a minimal non-abelian group, otherwise the output is false.
Recall that each finite non-abelian group whose proper subgroups are abelian is called a Miller-Moreno group or in other terminology a minimal non-abelian group.
gap> H:=SmallGroup(120,4); <pc group of size 120 with 5 generators> gap> IsMinimalNonAbelianGroup(H); false gap> K:=SmallGroup(16,6); <pc group of size 16 with 4 generators> gap> IsMinimalNonAbelianGroup(K); true gap> IsMinimalNonAbelianGroup(SmallGroup(16,8)); false
‣ IsMetacyclicPGroup( G ) | ( property ) |
Returns: a boolean
The argument is a group \(G\). The output is true if \(G\) is a metacyclic \(p\)-group, otherwise the output is false.
gap> IsMetacyclicPGroup(K); true gap> IsMetacyclicPGroup(SmallGroup(81,4)); true gap> IsMetacyclicPGroup(SmallGroup(81,15)); false
‣ EndoOrbitsOfGroup( G ) | ( operation ) |
Returns: EndoOrbitsOfGroup
The argument is a group \(G\). The output is
gap> D:=SmallGroup(81,2); <pc group of size 81 with 4 generators> gap> T:=EndoOrbitsOfGroup(D);; gap> Length(T); 1 gap> Size(T[1][2]); 81
‣ IsEndoCyclicGroup( G ) | ( property ) |
Returns: a boolean
The argument is a group \(G\). The output is true if \(G\) is a endocyclic group, otherwise the output is false.
Let \(G\) be a group and \(End G\) be the set of all its endomorphisms, which can be considered as a semigroup with respect to the composition operation of endomorphisms. For each \(g\in G\) we denote by \(g^{End G}\) the set \(\{g^\alpha| \alpha\in End G\}\) of all images of the element \(g\) with respect to endomorphisms of \(End G\). A group \(G\) is called endocyclic if it contains an element \(g\) with \(G=g^{End G}\).
gap> IsEndoCyclicGroup(D); true
‣ UnitsOfNearRing( R ) | ( attribute ) |
Returns: a set
The argument is a nearring \(R\). The output is true if \(R\) is a nearring with identity, otherwise the output is Error, no units exist.
gap> N:=LocalNearRing(32,5,16,3,8); ExplicitMultiplicationNearRing ( <pc group of size 32 with 5 generators> , multiplication ) gap> U:=UnitsOfNearRing(N); [ (f1), (f1*f5), (f1*f4), (f1*f4*f5), (f1*f3), (f1*f3*f5), (f1*f3*f4), (f1*f3*f4*f5), (f1*f2), (f1*f2*f5), (f1*f2*f4), (f1*f2*f4*f5), (f1*f2*f3), (f1*f2*f3*f5), (f1*f2*f3*f4), (f1*f2*f3*f4*f5) ] gap> Un:=NearRingUnits(N);; U=Un; true
‣ IsLocalNearRing( R ) | ( property ) |
Returns: a boolean
The argument is a nearring \(R\). The output is true if \(R\) is a local nearring, otherwise the output is false.
gap> H:=SmallGroup(16,6); <pc group of size 16 with 4 generators> gap> A:= AutomorphismNearRing(H); AutomorphismNearRing( <pc group of size 16 with 4 generators> ) gap> Size(A); 64 gap> IsLocalNearRing(A); true gap> K:=LibraryNearRingWithOne(SmallGroup(8,2),1); #I using isomorphic copy of the group LibraryNearRing(8/2, 814) gap> IsLocalNearRing(K); false
‣ IsLocalRing( R ) | ( property ) |
Returns: a boolean
The argument is a local nearring \(R\). The output is true if \(R\) is a local ring, otherwise the output is false.
gap> L:=AllLocalNearRings(16,14,8,4);; gap> Size(L); 24 gap> F:=Filtered(L,x->IsLocalRing(x));; gap> Size(F); 1
‣ NearRingNonUnits( R ) | ( attribute ) |
Returns: a set
The argument is a nearring \(R\). The output is the set of non-invertible elements of \(R\).
gap> T:=LocalNearRing(49,2,42,1,1); ExplicitMultiplicationNearRing ( <pc group of size 49 with 2 generators> , multiplication ) gap> Nu:=NearRingNonUnits(T); [ (<identity> of ...), (f2), (f2^2), (f2^3), (f2^4), (f2^5), (f2^6) ] gap> Size(Nu); 7 gap> R:=LibraryNearRing(SmallGroup(8,4),3); #I using isomorphic copy of the group LibraryNearRing(8/5, 3) gap> N:=NearRingNonUnits(R); [ (()), ((1,2,3,4)(5,6,7,8)), ((1,3)(2,4)(5,7)(6,8)), ((1,4,3,2)(5,8,7,6)), ((1,5,3,7)(2,8,4,6)), ((1,6,3,8)(2,5,4,7)), ((1,7,3,5)(2,6,4,8)), ((1,8,3,6)(2,7,4,5)) ]
‣ SubNearRingByGenerators( R, gens ) | ( operation ) |
Returns: a subnearring
The arguments are a nearring \(R\) and generators \(gens\) of \(R\). The output is the subnearring generated by \(gens\).
gap> B:=LocalNearRing(25,2,20,3,1); ExplicitMultiplicationNearRing ( <pc group of size 25 with 2 generators> , multiplication ) gap> D:=DistributiveElements(B);; gap> Size(D); 5 gap> Rs:=SubNearRingByGenerators(B,D);; gap> Size(Rs); 5 gap> IsDgNearRing(B); false gap> IsDgNearRing(Rs); true
‣ NonUnitsAsAdditiveSubgroup( R ) | ( attribute ) |
Returns: a subgroup
The argument is a local nearring \(R\). The output is the additive subgroup of non-units of \(R\).
gap> T:=LocalNearRing(125,4,100,9,1); ExplicitMultiplicationNearRing ( <pc group of size 125 with 3 generators> , multiplication ) gap> L:=NonUnitsAsAdditiveSubgroup(T); Group([ <identity> of ..., f2, f3, f2^2, f2*f3, f3^2, f2^3, f2^2*f3, f2*f3^2, f3^3, f2^4, f2^3*f3, f2^2*f3^2, f2*f3^3, f3^4, f2^4*f3, f2^3*f3^2, f2^2*f3^3, f2*f3^4, f2^4*f3^2, f2^3*f3^3, f2^2*f3^4, f2^4*f3^3, f2^3*f3^4, f2^4*f3^4 ]) gap> IdGroup(L); [ 25, 2 ]
‣ NonUnitsAsNearRingIdeal( R ) | ( attribute ) |
Returns: an ideal
The argument is a local nearring \(R\). The output is the ideal generated by all non-invertible elements of \(R\).
gap> I:=NonUnitsAsNearRingIdeal(T); < nearring ideal > gap> Size(I); 25
‣ MultiplicativeSemigroupOfNearRing( R ) | ( attribute ) |
Returns: a semigroup
The argument is a nearring \(R\). The output is the multiplicative semigroup of \(R\).
gap> B:=LocalNearRing(16,10,8,2,7);; gap> M:=MultiplicativeSemigroupOfNearRing(B); Semigroup with identity (f1) <semigroup of size 16, with 7 generators> gap> Size(M); 16
‣ NonUnitsAsMultiplicativeSemigroup( R ) | ( attribute ) |
Returns: a semigroup
The argument is a nearring \(R\). The output is the multiplicative semigroup of non-units of \(R\).
gap> Nm:=NonUnitsAsMultiplicativeSemigroup(B); <semigroup with 8 generators> gap> Size(Nm); 8
‣ IsOneGeneratedNearRing( R ) | ( property ) |
Returns: a boolean
The argument is a nearring \(R\). The output is true if \(R\) is a nearring generated by one element, otherwise the output is false.
gap> D:=LocalNearRing(49,2,42,4,1); ExplicitMultiplicationNearRing ( <pc group of size 49 with 2 generators> , multiplication ) gap> IsOneGeneratedNearRing(D); true gap> H:=LocalNearRing(16,14,8,2,3); ExplicitMultiplicationNearRing ( <pc group of size 16 with 4 generators> , multiplication ) gap> IsOneGeneratedNearRing(H); false
‣ AutomorphismsAssociatedWithNearRingUnits( R, Un ) | ( operation ) |
Returns: automorphisms
The arguments are a nearring \(R\) with identity and a set of units \(Un\) of \(R\). The output are the automorphisms associated with nearring units.
A subgroup \(A\) of the automorphism group \(Aut R^+\) of the additive group of the nearring \(R\) with identity isomorphic to the multiplicative group \(R^*\) and satisfies the condition
\[i^A=\{i^a\mid a\in A\}=R^*\]
is called the subgroup of \(Aut R^+\) associated with the group \(R^*\).
gap> S:=UnitsOfNearRing(D); [ (f1), (f1*f2), (f1*f2^2), (f1*f2^3), (f1*f2^4), (f1*f2^5), (f1*f2^6), (f1^2), (f1^2*f2), (f1^2*f2^2), (f1^2*f2^3), (f1^2*f2^4), (f1^2*f2^5), (f1^2*f2^6), (f1^3), (f1^3*f2), (f1^3*f2^2), (f1^3*f2^3), (f1^3*f2^4), (f1^3*f2^5), (f1^3*f2^6), (f1^4), (f1^4*f2), (f1^4*f2^2), (f1^4*f2^3), (f1^4*f2^4), (f1^4*f2^5), (f1^4*f2^6), (f1^5), (f1^5*f2), (f1^5*f2^2), (f1^5*f2^3), (f1^5*f2^4), (f1^5*f2^5), (f1^5*f2^6), (f1^6), (f1^6*f2), (f1^6*f2^2), (f1^6*f2^3), (f1^6*f2^4), (f1^6*f2^5), (f1^6*f2^6) ] gap> A:=AutomorphismsAssociatedWithNearRingUnits(D,S);; gap> Size(A); 42
‣ EndomorphismsAssociatedWithNearRingElements( R, Elm ) | ( operation ) |
Returns: endomorphisms
The arguments are a nearring \(R\) and a set \(Elm\) of nearring elements. The output is the endomorphisms associated with nearring elements.
gap> Nu:=NearRingNonUnits(D); [ (<identity> of ...), (f2), (f2^2), (f2^3), (f2^4), (f2^5), (f2^6) ] gap> En:=EndomorphismsAssociatedWithNearRingElements(D,Nu);; gap> Size(En); 7
‣ SemidirectProductAssociatedWithNearRing( R ) | ( operation ) |
Returns: a semidirect product
The argument is a nearring \(R\) with identity. The output is the semidirect product associated with nearring \(R\).
gap> T:=LocalNearRing(25,2,20,2,1); ExplicitMultiplicationNearRing ( <pc group of size 25 with 2 generators> , multiplication ) gap> SemidirectProductAssociatedWithNearRing(T); <pc group with 5 generators> gap> Size(last); 500
‣ IsCircleSubgroupOfNearRing( R, H ) | ( operation ) |
Returns: a boolean
The arguments are a nearring \(R\) with identity and a subgroup \(H\) of additive group of \(R\). The output is true if \(H\) is a constructive subgroup of nearring \(R\), otherwise the output is false.
gap> Sg:=Subgroups(GroupReduct(T));; gap> Size(Sg); 8 gap> F:=Filtered(Sg,x->IsCircleSubgroupOfNearRing(T,x)); [ Group([ ]), Group([ f2 ]) ]
‣ FactorizedGroupAssociatedWithCircleSubgroupOfNearRing( R, H ) | ( operation ) |
Returns: a group
The arguments are a nearring \(R\) with identity and a constructive subgroup \(H\) of \(R\). The output is the group
gap> FG:=FactorizedGroupAssociatedWithCircleSubgroupOfNearRing(T,F[2]); <pc group with 2 generators> gap> IdGroup(FG); [ 25, 2 ]
‣ ConstantPartOfNearRing( R ) | ( attribute ) |
Returns: a constant part
The argument is a nearring \(R\). The output is the constant part of nearring \(R\).
gap> H:=LocalNearRing(361,2,342,7,7); ExplicitMultiplicationNearRing ( <pc group of size 361 with 2 generators> , multiplication ) gap> C:=ConstantPartOfNearRing(H);; gap> Size(C); 19
‣ ZeroSymmetricPartOfNearRing( R ) | ( attribute ) |
Returns: a zero-symmetric part
The argument is a nearring \(R\). The output is the zero-symmetric part of nearring \(R\).
gap> ZeroSymmetricPartOfNearRing(H);; gap> Size(last); 19
‣ GroupOfUnitsAsGroupOfAutomorphisms( R ) | ( attribute ) |
Returns: a group of units
The argument is a nearring \(R\). The output is the group of units as group of automorphisms \(R\).
gap> M:=LocalNearRing(27,4,18,3,2); ExplicitMultiplicationNearRing ( <pc group of size 27 with 3 generators> , multiplication ) gap> GroupOfUnitsAsGroupOfAutomorphisms(M); <group of size 18 with 2 generators> gap> Size(last); 18
‣ IsDistributiveElementOfNearRing( R, r ) | ( operation ) |
Returns: a boolean
The argument is a nearring \(R\) and an element \(r\). The output is true if \(r\) is a distributive element of nearring \(R\), otherwise the output is false.
gap> D:=LocalNearRing(49,2,42,6,1); ExplicitMultiplicationNearRing ( <pc group of size 49 with 2 generators> , multiplication ) gap> h:=List(D);; gap> d:=h[3]; (f2^2) gap> IsDistributiveElementOfNearRing(D,d); true
‣ IsSemiDistributiveNearRing( R ) | ( property ) |
Returns: a boolean
The argument is a nearring \(R\). The output is true if \(R\) is a semidistributive nearring, otherwise the output is false.
gap> N:=LocalNearRing(16,10,8,2,7); ExplicitMultiplicationNearRing ( <pc group of size 16 with 4 generators> , multiplication ) gap> IsSemiDistributiveNearRing(N); true
‣ IsNearRingWithIdentity( R ) | ( property ) |
Returns: a boolean
The argument is a nearring \(R\). The output is true if \(R\) is a nearring with identity, otherwise the output is false.
gap> N:=LocalNearRing(343,5,294,8,2); ExplicitMultiplicationNearRing ( <pc group of size 343 with 3 generators> , multiplication ) gap> IsNearRingWithOne(N); false gap> Identity(N); (f1) gap> IsNearRingWithIdentity(N); true
‣ IsSubNearRing( R, H ) | ( operation ) |
Returns: a boolean
The arguments are a nearring \(R\) with identity and a subgroup \(H\) of the additive group of \(R\). The output is true if \(H\) is the additive group of a subnearring of \(R\), otherwise the output is false.
gap> T:=LocalNearRing(49,2,42,1,2); ExplicitMultiplicationNearRing ( <pc group of size 49 with 2 generators> , multiplication ) gap> G:=GroupReduct(T); <pc group of size 49 with 2 generators> gap> S:=Subgroups(G);; gap> Size(S); 10 gap> IsSubNearRing(T,S[3]); true gap> IsSubNearRing(T,S[9]); false gap> D:=SmallGroup(7,1); <pc group of size 7 with 1 generators> gap> IsSubNearRing(T,D); false
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