GENERIC ABELIAN GROUPS

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GENERIC ABELIAN GROUPS

Introduction

Construction of a Generic Abelian Group

Elements of a Generic Abelian Group
Constructing an Element of a Generic Abelian Group
Pull-Back of an Element
Representation of an Element

Structure Computation

Subgroups
Construction of Subgroups
Construction of p-Sylow Subgroups

Access Functions

Arithmetic with Elements
Addition and Subtraction

Operations on Elements
Order and Discrete Logarithm
Equality and Comparison

Set-Theoretic Operations
Membership and Equality

Homomorphisms

Bibliography

DETAILS

Introduction

Construction of a Generic Abelian Group
GenericAbelianGroup(U: parameters) : . -> GrpAbGen
Example GrpAbGen_Creation (H17E1)

Elements of a Generic Abelian Group

Constructing an Element of a Generic Abelian Group
A ! e : GrpAbGen, Elt -> GrpAbGenElt
A ! g : GrpAbGen, GrpAbGenElt -> GrpAbGenElt
A ! [a_1, ... ,a_n] : GrpAbGen, [RngIntElt] -> GrpAbGenElt
Identity(A) : GrpAbGen -> GrpAbGenElt
Random(A) : GrpAbGen -> GrpAbGenElt

Pull-Back of an Element

Representation of an Element
Representation(g) : GrpAbGenElt -> [RngIntElt]
UserRepresentation(g) : GrpAbGenElt -> [RngIntElt]
Representation(S, g) : SeqEnum, GrpAbGenElt -> [RngIntElt], RngIntElt
Example GrpAbGen_ElementCreationAndRep (H17E2)

Structure Computation
AbelianGroup(A: parameters) : GrpAbGen -> GrpAb, Map
Example GrpAbGen_GroupComputation (H17E3)

Construction of Subgroups
sub<A | L: parameters> : GrpAbGen, List -> GrpAbGen
Example GrpAbGen_SubgroupCreation (H17E4)

Construction of p-Sylow Subgroups
Sylow(A, p: parameters) : GrpAbGen, RngInt -> GrpAbGen
Example GrpAbGen_pSylowComputation (H17E5)

Access Functions
Universe(A) : GrpAbGen ->
Order(A) : GrpAbGen -> RngIntElt
A . i : GrpAbGen, RngIntElt -> GrpAbGenElt
Generators(A) : GrpAbGen -> [ GrpAbGenElt ]
UserGenerators(A) : GrpAbGen -> [ GrpAbGenElt ]
NumberOfGenerators(A) : GrpAbGen -> RngIntElt
Invariants(A) : GrpAbGen -> [ RngIntElt ]
Example GrpAbGen_ElementCreationAndRep (H17E6)

Arithmetic with Elements

Addition and Subtraction
g + d : GrpAbGenElt, GrpAbGenElt -> GrpAbGenElt
- g : GrpAbGenElt -> GrpAbGenElt
g - d : GrpAbGenElt, GrpAbGenElt -> GrpAbGenElt
n * g : RngIntElt, GrpAbGenElt-> GrpAbGenElt

Operations on Elements

Order and Discrete Logarithm
Order(g: parameters) : GrpAbGenElt -> RngIntElt
Order(g, l, u: parameters) : GrpAbGenElt, RngIntElt, RngIntElt -> RngIntElt
Order(g, l, u, n, m: parameters) : GrpAbGenElt, RngIntElt, RngIntElt ,RngIntElt, RngIntElt -> RngIntElt
Log(g, d: parameters) : GrpAbGenElt, GrpAbGenElt -> RngIntElt
Example GrpAbGen_ElementOrderLog (H17E7)

Equality and Comparison
g eq d : GrpAbGenElt, GrpAbGenElt -> BoolElt
g ne d : GrpAbGenElt, GrpAbGenElt -> BoolElt
IsIdentity(g) : GrpAbGenElt -> BoolElt

Set-Theoretic Operations

Membership and Equality
g in A : GrpAbGenElt, GrpAbGen -> BoolElt
g notin A : GrpAbGenElt, GrpAbGen -> BoolElt
S subset A : { GrpAbGenElt } , GrpAbGen -> BoolElt
S notsubset A : { GrpAbGenElt } , GrpAbGen -> BoolElt
H subset A : GrpAbGen, GrpAbGen -> BoolElt
H notsubset A : GrpAbGen, GrpAbGen -> BoolElt
A eq B : GrpAbGen, GrpAbGen -> BoolElt
A ne B : GrpAbGen, GrpAbGen -> BoolElt

Homomorphisms
hom< A -> B | L> : Grp, Grp, List -> Map
Homomorphism(A, B, gens, images) : Grp, Grp, [ GrpElt ], [ GrpElt ] -> Map

Bibliography