Interface AbelianGroup<E>

All Superinterfaces:
AbelianMonoid<E>, Group<E>, Magma<E>, Monoid<E>, Set<E>
All Known Subinterfaces:
Algebra<E,F>, BanachAlgebra<E,F>, BanachSpace<E,S>, CStarAlgebra<E,F>, DivisionRing<E>, Field<E>, GradedAlgebra<E,F>, HilbertSpace<E,S>, LieAlgebra<E,S>, Matrix<E>, Module<M,R>, NonAssociativeAlgebra<E,S>, Ring<E>, Vector<E>, VectorSpace<V,F>
All Known Implementing Classes:
CliffordAlgebra, Complex, Complexes, DenseMatrix, DenseVector, DoubleField, GenericMatrix, GenericVector, Integer, Integers, MatrixLieAlgebra, MMapMatrix, NativeRealBig, NativeRealDoubleMatrix, NativeRealDoubleVector, NativeRealFloatMatrix, Octonion, Octonions, PolynomialRing, Quaternion, Quaternions, Rational, Rationals, Real, RealBig, RealDouble, RealDoubleMatrix, RealDoubleVector, RealFloat, RealFloatVector, Reals, SIMDRealDoubleMatrix, SIMDRealFloatMatrix, SparseMatrix, SparseVector, SquareMatrices, TiledMatrix, Vector2D, Vector3D, Vector4D, VectorND, VectorSpace2D, VectorSpace3D
public interface AbelianGroup<E> extends Group<E>, AbelianMonoid<E>
An abelian group is a commutative group.

Abelian groups are named after Niels Henrik Abel and are fundamental in many areas of mathematics. The addition operation in most number systems forms an abelian group.

Mathematical Definition

An abelian group (G, +) satisfies all group axioms plus:

  • Commutativity: ∀ a, b ∈ G: a + b = b + a

Convention: Abelian groups are typically written with additive notation (+, 0, −a) rather than multiplicative (∗, e, a⁻¹).

Examples

  • (ℤ, +) - Integers with addition
  • (â„š, +) - Rationals with addition
  • (ℝ, +) - Reals with addition
  • (â„‚, +) - Complex numbers with addition
  • (ℝⁿ, +) - n-dimensional vectors with addition
  • (ℤ/nℤ, +) - Integers modulo n

Additive Notation

Since:
1.0
Author:
Silvere Martin-Michiellot, Gemini AI (Google DeepMind)