Interface Group

All Known Subinterfaces:
DecompositionStore<N>, Field<T>, Group.Additive<T>, Group.Multiplicative<T>, Matrix2D<N,M>, MatrixStore<N>, NormedVectorSpace<T,N>, PhysicalStore<N>, PolynomialFunction<N>, Ring<T>, Scalar<N>, SelfDeclaringScalar<S>, Tensor<N,T>, VectorSpace<T,N>
All Known Implementing Classes:
AboveBelowStore, AbstractPolynomial, AbstractStore, Amount, AnyTensor, ArrayBasedTensor, BasicMatrix, BigScalar, ColumnsStore, ColumnsSupplier, ComplexNumber, ComposingStore, CompressedSparseR064, ConjugatedStore, DiagonalStore, ExactDecimal, FactoryStore, GenericStore, IdentityStore, ImageData, ImageData.SingleChannel, LeftRightStore, LimitStore, LogicalStore, LowerHessenbergStore, LowerSymmetricStore, LowerTriangularStore, MatrixC128, MatrixH256, MatrixQ128, MatrixR032, MatrixR064, MatrixR128, MatrixTensor, Money, OffsetStore, PolynomialC128, PolynomialQ128, PolynomialR032, PolynomialR064, PolynomialR128, PolynomialR256, Price, PrimitiveScalar, Quadruple, Quantity, Quaternion, R032Store, R064CSC, R064CSR, R064Store, RationalNumber, RawStore, RepeatedColumnsStore, RepeatedRowsStore, RowsStore, RowsSupplier, ScalarPolynomial, SelectingStore, ShadingStore, SingleStore, SparseStore, SuperimposedStore, TransjugatedStore, TransposedStore, UnaryOperatoStore, UpperHessenbergStore, UpperSymmetricStore, UpperTriangularStore, VectorTensor, WrapperStore, ZeroStore
public interface Group

A group is a set of elements paired with a binary operation. Four conditions called the group axioms must be satisfied:

  • Closure: If A and B are both members of the set then the result of A op B is also a member.
  • Associativity: Invocation/execution order doesn't matter - ((A op B) op C) == (A op (B op C))
  • The identity property: There is an identity element in the set, I, so that I op A == A op I == A
  • The inverse property: For each element in the set there must be an inverse element (opposite or reciprocal) so that A-1 op A == A op A-1 == I

Note that commutativity is not a requirement - A op B doesn't always have to be equal to B op A. If the operation is commutative then the group is called an abelian group or simply a commutative group.

See Also: