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In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group.
A group G is said to be linear if there exists a field K, an integer d and an injective homomorphism from G to the general linear group GL d (K) (a faithful linear representation of dimension d over K): if needed one can mention the field and dimension by saying that G is linear of degree d over K.
Conjugation is an action by automorphisms, i.e., it commutes with the group structure, and this induces linear actions on naturally derived objects, such as its Lie algebra, and the algebra of left-invariant differential operators. An S-group scheme G is commutative if the group G(T) is an abelian group for all S-schemes T. There are several ...
The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1. In the simple case n = 1, the group U(1) corresponds to the circle group, isomorphic to the set of all complex numbers that have absolute value 1, under multiplication ...
The general linear group GL(2, 7) consists of all invertible 2×2 matrices over F 7, the finite field with 7 elements. These have nonzero determinant. The subgroup SL(2, 7) consists of all such matrices with unit determinant.
A fundamental example of a reductive group is the general linear group of invertible n × n matrices over a field k, for a natural number n. In particular, the multiplicative group G m is the group GL (1), and so its group G m ( k ) of k -rational points is the group k * of nonzero elements of k under multiplication.
A noteworthy subgroup of the projective general linear group PGL(2, Z) (and of the projective special linear group PSL(2, Z[i])) is the symmetries of the set {0, 1, ∞} ⊂ P 1 (C) [note 6] which is known as the anharmonic group, and arises as the symmetries of the six cross-ratios.
The general linear group GL n (R) is the group of all R-linear automorphisms of R n. There is a subgroup: the special linear group SL n (R), and their quotients: the projective general linear group PGL n (R) = GL n (R)/Z(GL n (R)) and the projective special linear group PSL n (R) = SL n (R)/Z(SL n (R)).