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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.
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 ...
General linear group; General semilinear group; GL(n,C) Group algebra of a locally compact group; Group analysis of differential equations; Group contraction; H. Haar ...
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.
The quotient G/Φ(G) is an elementary abelian group and its automorphism group is a general linear group, so very well understood. The map from the automorphism group of G into this general linear group has been studied by Burnside, who showed that the kernel of this map is a p-group.
In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the general linear group GL n (n x n invertible matrices), the subgroup of invertible upper triangular matrices is a Borel subgroup.
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)).