The Free Abelian Group ：

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A<[x]> := AbelianGroup([2,3,4,5,6,10,0]); > A; A . 1; A . 6 ; Generators(A) ; NumberOfGenerators(A); Ngens(A); Parent(40); Parent(3); Relations(A); RelationMatrix(A) ; r:= 10=3; r; A; Generators(A) ; RelationMatrix(A) ; Ar:=AbelianGroup< A | r > ; Ar; Generators(Ar) ; RelationMatrix(Ar) ; r[1] ; r[2] ; LHS(r); RHS(r) ; Parent(r); UserGenerators(A) ; UserGenerators(Ar); As:=sub< A | 3*x[1], 333*x[3], x[5], x[7] >; As; Aq:=A/As; Aq; Generators(Aq) ; RelationMatrix(Aq) ; Invariants(A) ; Invariants(Ar) ; Invariants(Aq) ; TorsionFreeRank(A) ; TorsionFreeRank(Ar) ; TorsionFreeRank(Aq) ; TorsionInvariants(A) ; TorsionInvariants(Ar) ; TorsionInvariants(Aq) ; PrimaryInvariants(A); PrimaryInvariants(Ar); PrimaryInvariants(Aq); pPrimaryInvariants(A, 3); pPrimaryInvariants(Ar, 2); pPrimaryInvariants(Aq, 3); TorsionFreeSubgroup(A) ; TorsionFreeSubgroup(Ar) ; TorsionFreeSubgroup(Aq) ; TorsionSubgroup(A); TorsionSubgroup(Ar); TorsionSubgroup(Aq); pPrimaryComponent(A, 2); pPrimaryComponent(Ar, 3); pPrimaryComponent(Aq, 5); Abelian Group isomorphic to Z/2 + Z/2 + Z/30 + Z/60 + Z Defined on 7 generators Relations: 2*x[1] = 0 3*x[2] = 0 4*x[3] = 0 5*x[4] = 0 6*x[5] = 0 10*x[6] = 0 x[1] x[6] { x[3], x[7], x[1], x[2], x[4], x[6], x[5] } 7 7 Integer Ring Integer Ring [ 2*$.1 = 0, 3*$.2 = 0, 4*$.3 = 0, 5*$.4 = 0, 6*$.5 = 0, 10*$.6 = 0 ] [ 2 0 0 0 0 0 0] [ 0 3 0 0 0 0 0] [ 0 0 4 0 0 0 0] [ 0 0 0 5 0 0 0] [ 0 0 0 0 6 0 0] [ 0 0 0 0 0 10 0] 10 = 3 Abelian Group isomorphic to Z/2 + Z/2 + Z/30 + Z/60 + Z Defined on 7 generators Relations: 2*x[1] = 0 3*x[2] = 0 4*x[3] = 0 5*x[4] = 0 6*x[5] = 0 10*x[6] = 0 { x[3], x[7], x[1], x[2], x[4], x[6], x[5] } [ 2 0 0 0 0 0 0] [ 0 3 0 0 0 0 0] [ 0 0 4 0 0 0 0] [ 0 0 0 5 0 0 0] [ 0 0 0 0 6 0 0] [ 0 0 0 0 0 10 0] Abelian Group isomorphic to Z/7 Defined on 1 generator Relations: 7*Ar.1 = 0 { Ar.1 } [7] 10 3 10 3 RelationWorld [ x[1], x[2], x[3], x[4], x[5], x[6], x[7] ] [ Ar.1 ] Abelian Group isomorphic to Z/2 + Z/2 + Z/12 + Z Defined on 4 generators in supergroup A: As.1 = x[1] As.2 = x[1] + 2*x[3] + 3*x[5] As.3 = x[1] + x[3] + 5*x[5] As.4 = x[7] Relations: 2*As.1 = 0 2*As.2 = 0 12*As.3 = 0 Abelian Group isomorphic to Z/5 + Z/30 Defined on 2 generators Relations: 5*Aq.1 = 0 30*Aq.2 = 0 { Aq.2, Aq.1 } [ 5 0] [ 0 30] [ 2, 2, 30, 60, 0 ] [ 7 ] [ 5, 30 ] 1 0 0 [ 2, 2, 30, 60 ] [ 7 ] [ 5, 30 ] [ 2, 2, 2, 4, 3, 3, 5, 5 ] [ 7 ] [ 2, 3, 5, 5 ] [ 3, 3 ] [] [ 3 ] Abelian Group isomorphic to Z Defined on 1 generator in supergroup A: $.1 = x[7] (free) Abelian Group of order 1 Abelian Group of order 1 Abelian Group isomorphic to Z/2 + Z/2 + Z/30 + Z/60 Defined on 4 generators in supergroup A: $.1 = 2*x[3] + 5*x[6] $.2 = x[1] $.3 = x[1] + 4*x[4] + x[5] $.4 = 2*x[2] + 3*x[3] + x[6] Relations: 2*$.1 = 0 2*$.2 = 0 30*$.3 = 0 60*$.4 = 0 Abelian Group isomorphic to Z/7 Defined on 1 generator Relations: 7*Ar.1 = 0 Abelian Group isomorphic to Z/5 + Z/30 Defined on 2 generators Relations: 5*Aq.1 = 0 30*Aq.2 = 0 Abelian Group isomorphic to Z/2 + Z/2 + Z/2 + Z/4 Defined on 4 generators in supergroup A: $.1 = 2*x[3] + 5*x[6] $.2 = x[1] $.3 = x[1] + 3*x[5] $.4 = x[3] + 5*x[6] Relations: 2*$.1 = 0 2*$.2 = 0 2*$.3 = 0 4*$.4 = 0 Abelian Group of order 1 Abelian Group isomorphic to Z/5 + Z/5 Defined on 2 generators in supergroup Aq: $.1 = Aq.1 $.2 = 6*Aq.2 Relations: 5*$.1 = 0 5*$.2 = 0