  
  [1X2 [33X[0;0YFunctions[133X[101X
  
  
  [1X2.1 [33X[0;0YGroup functions[133X[101X
  
  [1X2.1-1 IsMinimalNonAbelianGroup[101X
  
  [33X[1;0Y[29X[2XIsMinimalNonAbelianGroup[102X( [3XG[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument is a group [23XG[123X. The output is [10Xtrue[110X if [23XG[123X is a minimal non-abelian
  group, otherwise the output is [10Xfalse[110X.[133X
  
  [33X[0;0YRecall that each finite non-abelian group whose proper subgroups are abelian
  is  called  a  [13XMiller-Moreno  group[113X  or,  in  other  terminology,  a [13Xminimal
  non-abelian group[113X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XH:=SmallGroup(120,4);       [127X[104X
    [4X[28X<pc group of size 120 with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsMinimalNonAbelianGroup(H);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XK:=SmallGroup(16,6);[127X[104X
    [4X[28X<pc group of size 16 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsMinimalNonAbelianGroup(K);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsMinimalNonAbelianGroup(SmallGroup(16,8));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X2.1-2 IsMetacyclicPGroup[101X
  
  [33X[1;0Y[29X[2XIsMetacyclicPGroup[102X( [3XG[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument is a group [23XG[123X. The output is [10Xtrue[110X if [23XG[123X is a metacyclic [23Xp[123X-group,
  otherwise the output is [10Xfalse[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsMetacyclicPGroup(K);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsMetacyclicPGroup(SmallGroup(81,4));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsMetacyclicPGroup(SmallGroup(81,15));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X2.1-3 EndoOrbitsOfGroup[101X
  
  [33X[1;0Y[29X[2XEndoOrbitsOfGroup[102X( [3XG[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YEndoOrbitsOfGroup[133X
  
  [33X[0;0YThe argument is a group [23XG[123X. The output is a list of pairs [10X[x,H][110X, where [10Xx[110X is a
  representative  of  an  endo-orbit  of [23XG[123X and [10XH[110X is the set of all images of [10Xx[110X
  under endomorphisms of [23XG[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD:=SmallGroup(81,2); [127X[104X
    [4X[28X<pc group of size 81 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XT:=EndoOrbitsOfGroup(D);;[127X[104X
    [4X[25Xgap>[125X [27XLength(T);[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XSize(T[1][2]);[127X[104X
    [4X[28X81[128X[104X
  [4X[32X[104X
  
  [1X2.1-4 IsEndoCyclicGroup[101X
  
  [33X[1;0Y[29X[2XIsEndoCyclicGroup[102X( [3XG[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is a group [23XG[123X. The output is [10Xtrue[110X if [23XG[123X is an endocyclic group,
  otherwise the output is [10Xfalse[110X.[133X
  
  [33X[0;0YLet [23XG[123X be a group and [23XEnd G[123X be the set of all its endomorphisms, which can be
  considered  as  a  semigroup with respect to composition. For each [23Xg\in G[123X we
  denote  by  [23Xg^{End G}[123X the set [23X\{g^\alpha| \alpha\in End G\}[123X of all images of
  the  element  [23Xg[123X  with respect to endomorphisms of [23XEnd G[123X. A group [23XG[123X is called
  [13Xendocyclic[113X if it contains an element [23Xg[123X with [23XG=g^{End G}[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsEndoCyclicGroup(D);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X2.2 [33X[0;0YNearring functions[133X[101X
  
  [1X2.2-1 UnitsOfNearRing[101X
  
  [33X[1;0Y[29X[2XUnitsOfNearRing[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YThe  argument  is  a  nearring  [23XR[123X.  The  output is a list of units if [23XR[123X is a
  nearring with identity, otherwise the output is an empty list.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XN:=LocalNearRing(32,5,16,3,8);[127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 32 with [128X[104X
    [4X[28X5 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XU:=UnitsOfNearRing(N);[127X[104X
    [4X[28X[ (f1), (f1*f5), (f1*f4), (f1*f4*f5), (f1*f3), (f1*f3*f5), (f1*f3*f4),  [128X[104X
    [4X[28X (f1*f3*f4*f5), (f1*f2), (f1*f2*f5), (f1*f2*f4), (f1*f2*f4*f5), (f1*f2*f3), [128X[104X
    [4X[28X (f1*f2*f3*f5), (f1*f2*f3*f4), (f1*f2*f3*f4*f5) ][128X[104X
    [4X[25Xgap>[125X [27XUn:=NearRingUnits(N);;[127X[104X
    [4X[25Xgap>[125X [27XU=Un;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XL:=LibraryNearRing(SmallGroup(6,1),3);[127X[104X
    [4X[28X#I  using isomorphic copy of the group[128X[104X
    [4X[28XLibraryNearRing(6/2, 3)[128X[104X
    [4X[25Xgap>[125X [27XUnitsOfNearRing(L);[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [1X2.2-2 IsLocalNearRing[101X
  
  [33X[1;0Y[29X[2XIsLocalNearRing[102X( [3XR[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is a nearring [23XR[123X. The output is [10Xtrue[110X if [23XR[123X is a local nearring,
  otherwise the output is [10Xfalse[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XH:=SmallGroup(16,6);[127X[104X
    [4X[28X<pc group of size 16 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XA:= AutomorphismNearRing(H);[127X[104X
    [4X[28XAutomorphismNearRing( <pc group of size 16 with 4 generators> )[128X[104X
    [4X[25Xgap>[125X [27X Size(A);[127X[104X
    [4X[28X64[128X[104X
    [4X[25Xgap>[125X [27XIsLocalNearRing(A);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XK:=LibraryNearRingWithOne(SmallGroup(8,2),1);  [127X[104X
    [4X[28X#I  using isomorphic copy of the group[128X[104X
    [4X[28XLibraryNearRing(8/2, 814)[128X[104X
    [4X[25Xgap>[125X [27XIsLocalNearRing(K);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X2.2-3 IsLocalRing[101X
  
  [33X[1;0Y[29X[2XIsLocalRing[102X( [3XR[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe argument is a local nearring [23XR[123X. The output is [10Xtrue[110X if [23XR[123X is a local ring,
  otherwise the output is [10Xfalse[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:=AllLocalNearRings(16,14,8,4);;[127X[104X
    [4X[25Xgap>[125X [27XSize(L);[127X[104X
    [4X[28X24[128X[104X
    [4X[25Xgap>[125X [27XF:=Filtered(L,x->IsLocalRing(x));;[127X[104X
    [4X[25Xgap>[125X [27XSize(F);[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [1X2.2-4 NearRingNonUnits[101X
  
  [33X[1;0Y[29X[2XNearRingNonUnits[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya set[133X
  
  [33X[0;0YThe  argument  is  a  nearring  [23XR[123X.  The  output is the set of non-invertible
  elements of [23XR[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:=LocalNearRing(49,2,42,1,1); [127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 49 with [128X[104X
    [4X[28X2 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XNu:=NearRingNonUnits(T);[127X[104X
    [4X[28X[ (<identity> of ...), (f2), (f2^2), (f2^3), (f2^4), (f2^5), (f2^6) ][128X[104X
    [4X[25Xgap>[125X [27XSize(Nu);[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27XR:=LibraryNearRing(SmallGroup(8,4),3);[127X[104X
    [4X[28X#I  using isomorphic copy of the group[128X[104X
    [4X[28XLibraryNearRing(8/5, 3)[128X[104X
    [4X[25Xgap>[125X [27XN:=SortedList(NearRingNonUnits(R)); [127X[104X
    [4X[28X[ (()), ((1,2,3,4)(5,6,7,8)), ((1,3)(2,4)(5,7)(6,8)), ((1,4,3,2)(5,8,7,6)), [128X[104X
    [4X[28X  ((1,5,3,7)(2,8,4,6)), ((1,6,3,8)(2,5,4,7)), ((1,7,3,5)(2,6,4,8)), [128X[104X
    [4X[28X  ((1,8,3,6)(2,7,4,5)) ][128X[104X
  [4X[32X[104X
  
  [1X2.2-5 SubNearRingByGenerators[101X
  
  [33X[1;0Y[29X[2XSubNearRingByGenerators[102X( [3XR[103X, [3Xgens[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya subnearring[133X
  
  [33X[0;0YThe  arguments  are a nearring [23XR[123X and generators [23Xgens[123X of [23XR[123X. The output is the
  subnearring generated by [23Xgens[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XB:=LocalNearRing(25,2,20,3,1); [127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 25 with 2 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XD:=DistributiveElements(B);;[127X[104X
    [4X[25Xgap>[125X [27XSize(D);[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XRs:=SubNearRingByGenerators(B,D);;[127X[104X
    [4X[25Xgap>[125X [27XSize(Rs);[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XIsDgNearRing(B);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDgNearRing(Rs);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.2-6 NonUnitsAsAdditiveSubgroup[101X
  
  [33X[1;0Y[29X[2XNonUnitsAsAdditiveSubgroup[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya subgroup[133X
  
  [33X[0;0YThe  argument  is a local nearring [23XR[123X. The output is the additive subgroup of
  non-units of [23XR[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:=LocalNearRing(125,4,100,9,1); [127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 125 with [128X[104X
    [4X[28X3 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XL:=NonUnitsAsAdditiveSubgroup(T);[127X[104X
    [4X[28XGroup([ <identity> of ..., f2, f3, f2^2, f2*f3, f3^2, f2^3, f2^2*f3, f2*f3^2, f3^3, f2^4, [128X[104X
    [4X[28X  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, [128X[104X
    [4X[28X  f2^3*f3^3, f2^2*f3^4, f2^4*f3^3, f2^3*f3^4, f2^4*f3^4 ])[128X[104X
    [4X[25Xgap>[125X [27XIdGroup(L);[127X[104X
    [4X[28X[ 25, 2 ][128X[104X
  [4X[32X[104X
  
  [1X2.2-7 NonUnitsAsNearRingIdeal[101X
  
  [33X[1;0Y[29X[2XNonUnitsAsNearRingIdeal[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan ideal[133X
  
  [33X[0;0YThe argument is a local nearring [23XR[123X. The output is the ideal generated by all
  non-invertible elements of [23XR[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XI:=NonUnitsAsNearRingIdeal(T);[127X[104X
    [4X[28X< nearring ideal >[128X[104X
    [4X[25Xgap>[125X [27XSize(I);[127X[104X
    [4X[28X25[128X[104X
  [4X[32X[104X
  
  [1X2.2-8 MultiplicativeSemigroupOfNearRing[101X
  
  [33X[1;0Y[29X[2XMultiplicativeSemigroupOfNearRing[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya semigroup[133X
  
  [33X[0;0YThe  argument is a nearring [23XR[123X. The output is the multiplicative semigroup of
  [23XR[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XB:=LocalNearRing(16,10,8,2,7);;[127X[104X
    [4X[25Xgap>[125X [27XM:=MultiplicativeSemigroupOfNearRing(B);[127X[104X
    [4X[28X<semigroup of size 16, with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(M);[127X[104X
    [4X[28X16[128X[104X
  [4X[32X[104X
  
  [1X2.2-9 NonUnitsAsMultiplicativeSemigroup[101X
  
  [33X[1;0Y[29X[2XNonUnitsAsMultiplicativeSemigroup[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya semigroup[133X
  
  [33X[0;0YThe  argument is a nearring [23XR[123X. The output is the multiplicative semigroup of
  non-units of [23XR[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNm:=NonUnitsAsMultiplicativeSemigroup(B);[127X[104X
    [4X[28X<semigroup with 8 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(Nm);[127X[104X
    [4X[28X8[128X[104X
  [4X[32X[104X
  
  [1X2.2-10 IsOneGeneratedNearRing[101X
  
  [33X[1;0Y[29X[2XIsOneGeneratedNearRing[102X( [3XR[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a  nearring  [23XR[123X.  The  output  is  [10Xtrue[110X if [23XR[123X is a nearring
  generated by one element, otherwise the output is [10Xfalse[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD:=LocalNearRing(49,2,42,4,1);[127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 49 with 2 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XIsOneGeneratedNearRing(D);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XH:=LocalNearRing(16,14,8,2,3);   [127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 16 with [128X[104X
    [4X[28X4 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XIsOneGeneratedNearRing(H);    [127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X2.2-11 AutomorphismsAssociatedWithNearRingUnits[101X
  
  [33X[1;0Y[29X[2XAutomorphismsAssociatedWithNearRingUnits[102X( [3XR[103X, [3XUn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of automorphisms[133X
  
  [33X[0;0YThe arguments are a nearring [23XR[123X with identity and a set of units [23XUn[123X of [23XR[123X. The
  output  is  the  list of automorphisms of the additive group of [23XR[123X associated
  with the units in [23XUn[123X.[133X
  
  [33X[0;0YA  subgroup [23XA[123X of the automorphism group [23XAut R^+[123X of the additive group of the
  nearring  [23XR[123X  with  identity  isomorphic  to the multiplicative group [23XR^*[123X and
  satisfying the condition[133X
  
  
  [24X[33X[0;6Yi^A=\{i^a\mid a\in A\}=R^*[133X
  
  [124X
  
  [33X[0;0Yis called the subgroup of [23XAut R^+[123X associated with [23XR^*[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS:=UnitsOfNearRing(D);[127X[104X
    [4X[28X[ (f1), (f1*f2), (f1*f2^2), (f1*f2^3), (f1*f2^4), (f1*f2^5), (f1*f2^6), (f1^2), (f1^2*f2), [128X[104X
    [4X[28X  (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), [128X[104X
    [4X[28X  (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), [128X[104X
    [4X[28X  (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), [128X[104X
    [4X[28X  (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), [128X[104X
    [4X[28X  (f1^6*f2^2), (f1^6*f2^3), (f1^6*f2^4), (f1^6*f2^5), (f1^6*f2^6) ][128X[104X
    [4X[25Xgap>[125X [27XA:=AutomorphismsAssociatedWithNearRingUnits(D,S);;[127X[104X
    [4X[25Xgap>[125X [27XSize(A);[127X[104X
    [4X[28X42[128X[104X
  [4X[32X[104X
  
  [1X2.2-12 EndomorphismsAssociatedWithNearRingElements[101X
  
  [33X[1;0Y[29X[2XEndomorphismsAssociatedWithNearRingElements[102X( [3XR[103X, [3XElm[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of endomorphisms[133X
  
  [33X[0;0YThe  arguments  are  a  nearring  [23XR[123X  and a set [23XElm[123X of nearring elements. The
  output is the endomorphisms associated with nearring elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNu:=NearRingNonUnits(D);[127X[104X
    [4X[28X[ (<identity> of ...), (f2), (f2^2), (f2^3), (f2^4), (f2^5), (f2^6) ][128X[104X
    [4X[25Xgap>[125X [27XEn:=EndomorphismsAssociatedWithNearRingElements(D,Nu);;[127X[104X
    [4X[25Xgap>[125X [27XSize(En);[127X[104X
    [4X[28X7[128X[104X
  [4X[32X[104X
  
  [1X2.2-13 SemidirectProductAssociatedWithNearRing[101X
  
  [33X[1;0Y[29X[2XSemidirectProductAssociatedWithNearRing[102X( [3XR[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya semidirect product[133X
  
  [33X[0;0YThe  argument  is  a  nearring [23XR[123X with identity. The output is the semidirect
  product associated with the nearring [23XR[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:=LocalNearRing(25,2,20,2,1);             [127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 25 with 2 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XSemidirectProductAssociatedWithNearRing(T);[127X[104X
    [4X[28X<pc group with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(last);[127X[104X
    [4X[28X500[128X[104X
  [4X[32X[104X
  
  [1X2.2-14 IsCircleSubgroupOfNearRing[101X
  
  [33X[1;0Y[29X[2XIsCircleSubgroupOfNearRing[102X( [3XR[103X, [3XH[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  arguments  are  a  nearring  [23XR[123X  with  identity  and a subgroup [23XH[123X of the
  additive  group  of  [23XR[123X.  The output is [10Xtrue[110X if [23XH[123X is a circle subgroup of the
  nearring [23XR[123X, otherwise the output is [10Xfalse[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSg:=Subgroups(GroupReduct(T));;[127X[104X
    [4X[25Xgap>[125X [27XSize(Sg);[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XF:=Filtered(Sg,x->IsCircleSubgroupOfNearRing(T,x));[127X[104X
    [4X[28X[ Group([  ]), Group([ f2 ]) ][128X[104X
  [4X[32X[104X
  
  [1X2.2-15 FactorizedGroupAssociatedWithCircleSubgroupOfNearRing[101X
  
  [33X[1;0Y[29X[2XFactorizedGroupAssociatedWithCircleSubgroupOfNearRing[102X( [3XR[103X, [3XH[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya group[133X
  
  [33X[0;0YThe  arguments  are a nearring [23XR[123X with identity and a circle subgroup [23XH[123X of [23XR[123X.
  The output is the semidirect product associated with [23XH[123X and the automorphisms
  determined by the corresponding units.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFG:=FactorizedGroupAssociatedWithCircleSubgroupOfNearRing(T,F[2]);[127X[104X
    [4X[28X<pc group with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XIdGroup(FG);[127X[104X
    [4X[28X[ 25, 2 ][128X[104X
  [4X[32X[104X
  
  [1X2.2-16 ConstantPartOfNearRing[101X
  
  [33X[1;0Y[29X[2XConstantPartOfNearRing[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya constant part[133X
  
  [33X[0;0YThe  argument  is  a  nearring  [23XR[123X.  The  output  is the constant part of the
  nearring [23XR[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XH:=LocalNearRing(361,2,342,7,7);[127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 361 with [128X[104X
    [4X[28X2 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XC:=ConstantPartOfNearRing(H);;[127X[104X
    [4X[25Xgap>[125X [27XSize(C);[127X[104X
    [4X[28X19[128X[104X
  [4X[32X[104X
  
  [1X2.2-17 ZeroSymmetricPartOfNearRing[101X
  
  [33X[1;0Y[29X[2XZeroSymmetricPartOfNearRing[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya zero-symmetric part[133X
  
  [33X[0;0YThe  argument  is a nearring [23XR[123X. The output is the zero-symmetric part of the
  nearring [23XR[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZeroSymmetricPartOfNearRing(H);;[127X[104X
    [4X[25Xgap>[125X [27XSize(last);[127X[104X
    [4X[28X19[128X[104X
  [4X[32X[104X
  
  [1X2.2-18 GroupOfUnitsAsGroupOfAutomorphisms[101X
  
  [33X[1;0Y[29X[2XGroupOfUnitsAsGroupOfAutomorphisms[102X( [3XR[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya group[133X
  
  [33X[0;0YThe  argument is a nearring [23XR[123X. The output is a group of automorphisms of the
  additive group of [23XR[123X that is isomorphic to the group of units of [23XR[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XM:=LocalNearRing(27,4,18,3,2);  [127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 27 with 3 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XGroupOfUnitsAsGroupOfAutomorphisms(M);[127X[104X
    [4X[28X<group of size 18 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XSize(last);[127X[104X
    [4X[28X18                [128X[104X
  [4X[32X[104X
  
  [1X2.2-19 IsDistributiveElementOfNearRing[101X
  
  [33X[1;0Y[29X[2XIsDistributiveElementOfNearRing[102X( [3XR[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument is a nearring [23XR[123X and an element [23Xr[123X. The output is [10Xtrue[110X if [23Xr[123X is a
  distributive element of the nearring [23XR[123X, otherwise the output is [10Xfalse[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD:=LocalNearRing(49,2,42,6,1); [127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 49 with [128X[104X
    [4X[28X2 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27Xh:=List(D);;[127X[104X
    [4X[25Xgap>[125X [27Xd:=h[3];[127X[104X
    [4X[28X(f2^2)[128X[104X
    [4X[25Xgap>[125X [27XIsDistributiveElementOfNearRing(D,d);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.2-20 IsSemiDistributiveNearRing[101X
  
  [33X[1;0Y[29X[2XIsSemiDistributiveNearRing[102X( [3XR[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument is a nearring [23XR[123X. The output is [10Xtrue[110X if [23XR[123X is a semidistributive
  nearring, otherwise the output is [10Xfalse[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XN:=LocalNearRing(16,10,8,2,7); [127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 16 with 4 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XIsSemiDistributiveNearRing(N);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.2-21 IsNearRingWithIdentity[101X
  
  [33X[1;0Y[29X[2XIsNearRingWithIdentity[102X( [3XR[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  argument  is  a  nearring [23XR[123X. The output is [10Xtrue[110X if [23XR[123X is a nearring with
  identity, otherwise the output is [10Xfalse[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XN:=LocalNearRing(343,5,294,8,2);    [127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 343 with [128X[104X
    [4X[28X3 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XIsNearRingWithOne(N);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIdentity(N);[127X[104X
    [4X[28X(f1)[128X[104X
    [4X[25Xgap>[125X [27XIsNearRingWithIdentity(N);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.2-22 IsSubNearRing[101X
  
  [33X[1;0Y[29X[2XIsSubNearRing[102X( [3XR[103X, [3XH[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe  arguments  are  a  nearring  [23XR[123X  with  identity  and a subgroup [23XH[123X of the
  additive  group  of  [23XR[123X.  The  output is [10Xtrue[110X if [23XH[123X is the additive group of a
  subnearring of [23XR[123X, otherwise the output is [10Xfalse[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:=LocalNearRing(49,2,42,1,2);        [127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 49 with [128X[104X
    [4X[28X2 generators> , multiplication )[128X[104X
    [4X[25Xgap>[125X [27XG:=GroupReduct(T);                    [127X[104X
    [4X[28X<pc group of size 49 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XS:=Subgroups(G);;[127X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X10[128X[104X
    [4X[25Xgap>[125X [27XIsSubNearRing(T,S[3]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSubNearRing(T,S[9]); [127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XD:=SmallGroup(7,1);;[127X[104X
    [4X[25Xgap>[125X [27XIsSubNearRing(T,D);   [127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
