Special linear group

Linear groups are used in mathematics to the description of symmetries. The special linear group of degree n over a field (or more generally a commutative, unitary ring), the group of all matrices with coefficients whose determinant is 1; these are also called unimodular matrices. Group join is the matrix multiplication

If the context is clear that the body is the set of real or complex numbers, we also write.

Properties

The special linear group is a normal subgroup of the general linear group.

The factor group is isomorphic to, the unit group of ( for a body is equal without the 0). The proof is via the homomorphism with the determinant as a homomorphism.

Important subgroups of the special orthogonal group are for and for the special unitary group.

The special linear group over the field or is a Lie group on the dimension.

The specific groups are linear algebraic groups, since the condition that the determinant must be equal to 1, can be expressed by a polynomial equation in the coefficient matrix.

The special linear group includes all orientation-preserving and volume- preserving linear maps.