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The infinite general linear group or stable general linear group is the direct limit of the inclusions GL(n, F) → GL(n + 1, F) as the upper left block matrix. It is denoted by either GL( F ) or GL(∞, F ) , and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places.
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.
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 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 ...
Given the affine group of an affine space A, the stabilizer of a point p is isomorphic to the general linear group of the same dimension (so the stabilizer of a point in Aff(2, R) is isomorphic to GL(2, R)); formally, it is the general linear group of the vector space (A, p): recall that if one fixes a point, an affine space becomes a vector space.
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)).
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.
The group operations are given by multiplying the matrices from the groups with the vectors from K n. The general linear group GL(n, Z) acts on Z n by natural matrix action. The orbits of its action are classified by the greatest common divisor of coordinates of the vector in Z n.