# General linear group

The general linear group or the degree over a body is the group of all regular - matrices made with coefficients . Group linking is matrix multiplication . The name comes from generally linear or the English term " g eneral l inear group". ${\ displaystyle \ mathrm {GL} (n, K)}$${\ displaystyle \ mathrm {GL} _ {n} (K)}$${\ displaystyle n}$ ${\ displaystyle K}$ ${\ displaystyle n \ times n}$${\ displaystyle K}$${\ displaystyle \ mathrm {GL}}$

If the body is a finite body with a prime power , one also writes instead . If it is clear from the context that the body of the real or the complex numbers is taken as a basis, one writes or . ${\ displaystyle K}$ ${\ displaystyle \ mathbb {F} _ {q}}$ ${\ displaystyle q}$${\ displaystyle \ mathrm {GL} (n, q)}$${\ displaystyle \ mathrm {GL} (n, K)}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {C}}$${\ displaystyle \ mathrm {GL} (n)}$${\ displaystyle \ mathrm {GL} _ {n}}$

The general linear group and its subgroups are used in the representation of groups as well as in the study of symmetries .

Subgroups of the general linear group are called matrix groups .

## General linear group over a vector space

If a vector space is over a body  , one writes or for the group of all automorphisms of , i.e. all bijective linear mappings , with the execution of such mappings one after the other as a group connection. ${\ displaystyle V}$${\ displaystyle K}$${\ displaystyle \ mathrm {GL} (V)}$${\ displaystyle \ mathrm {Aut} (V)}$${\ displaystyle V}$ ${\ displaystyle V \ to V}$

If the finite dimension has are and isomorphic. For a given basis of vector space , each automorphism of can be represented by an invertible matrix. This creates an isomorphism from to . ${\ displaystyle V}$ ${\ displaystyle n}$${\ displaystyle \ mathrm {GL} (V)}$${\ displaystyle \ mathrm {GL} (n, K)}$${\ displaystyle V}$${\ displaystyle V}$${\ displaystyle n \ times n}$${\ displaystyle \ mathrm {GL} (V)}$${\ displaystyle \ mathrm {GL} (n, K)}$

For the group is not Abelian . For example, ${\ displaystyle n \ geq 2}$${\ displaystyle \ mathrm {GL} (n, K)}$ ${\ displaystyle n = 2}$

${\ displaystyle {\ begin {pmatrix} 1 & 0 \\ 1 & 1 \ end {pmatrix}} {\ begin {pmatrix} 1 & 1 \\ 0 & 1 \ end {pmatrix}} = {\ begin {pmatrix} 1 & 1 \\ 1 & 2 \ end {pmatrix }}}$

but

${\ displaystyle {\ begin {pmatrix} 1 & 1 \\ 0 & 1 \ end {pmatrix}} {\ begin {pmatrix} 1 & 0 \\ 1 & 1 \ end {pmatrix}} = {\ begin {pmatrix} 2 & 1 \\ 1 & 1 \ end {pmatrix }}}$.

The center of consists of the multiples of the identity matrix (with scalars off ). ${\ displaystyle \ mathrm {GL} (n, K)}$${\ displaystyle K \ setminus \ {0 \}}$

## Subgroups of GL (n, K)

Each subgroup of is called a matrix group or linear group . Some subgroups have special meaning. ${\ displaystyle \ mathrm {GL} (n, K)}$

• The subgroup of all diagonal matrices , whose diagonal elements are all not equal to 0, describes rescaling of the space.
• Diagonal matrices, in which all diagonal elements match and are not 0, describe centric extensions in geometry . The subset of these matrices is the center of . Only in the trivial case it is identical.${\ displaystyle \ mathrm {GL} (n, K)}$${\ displaystyle n = 1}$${\ displaystyle \ mathrm {GL} (n, K)}$
• The special linear group consists of all matrices with the determinant 1. is a normal divisor of ; and the factor group is isomorphic to , the unit group of (without the 0).${\ displaystyle \ mathrm {SL} (n, K)}$${\ displaystyle \ mathrm {SL} (n, K)}$${\ displaystyle \ mathrm {GL} (n, K)}$ ${\ displaystyle \ mathrm {GL} (n, K) / \ mathrm {SL} (n, K)}$${\ displaystyle K ^ {\ times}}$${\ displaystyle K}$
• The orthogonal group contains all orthogonal matrices .${\ displaystyle \ mathrm {O} (n, K)}$
For these matrices describe automorphisms of which contain the Euclidean norm and the scalar product , i.e. orthogonal mappings .${\ displaystyle K = \ mathbb {R}}$${\ displaystyle \ mathbb {R} ^ {n}}$
• The unitary group consists of all unitary matrices , that is, those matrices whose adjoint is equal to its inverse . More generally, the unitary group can be defined as a subgroup of the linear mappings in a Prehilbert space , just as the orthogonal group can be understood as a subgroup of the linear mappings in a Euclidean vector space .${\ displaystyle \ mathrm {U} (n, \ mathbb {C})}$
• The affine group AGL n (K) is a subgroup of .${\ displaystyle \ mathrm {GL} (n + 1, K)}$

## About the real and complex numbers

The general linear group over the body or is an algebraic group and thus in particular a Lie group over the body and has the dimension . ${\ displaystyle \ mathrm {GL} (n)}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {C}}$${\ displaystyle n ^ {2}}$

Proof:
${\ displaystyle \ mathrm {GL} (n)}$is a subset of the manifold of all matrices, which is a vector space of dimension . The determinant is a polynomial and therefore especially a continuous mapping . As the archetype of the open subset of is an open, not empty subset of and therefore also has the dimension .${\ displaystyle \ mathrm {Mat} _ {n} (K)}$${\ displaystyle n \ times n}$${\ displaystyle n ^ {2}}$${\ displaystyle \ mathrm {Mat} _ {n} (K) \ \ rightarrow \ K}$${\ displaystyle \ mathrm {GL} (n)}$${\ displaystyle K ^ {\ times}}$${\ displaystyle K}$${\ displaystyle \ mathrm {Mat} _ {n} (K)}$${\ displaystyle n ^ {2}}$

The Lie algebra zu is the general linear Lie algebra . This consists of all matrices with the commutator as a Lie bracket . ${\ displaystyle \ mathrm {GL} (n)}$ ${\ displaystyle {\ mathfrak {gl}} (n)}$${\ displaystyle n \ times n}$

While connected is has two connected components: the matrices with positive and negative determinant. The connected component with a positive determinant contains the unit element and forms a subgroup . This subgroup is a connected Lie group with real dimensions and has the same Lie algebra as . ${\ displaystyle \ mathrm {GL} (n, \ mathbb {C})}$ ${\ displaystyle \ mathrm {GL} (n, \ mathbb {R})}$${\ displaystyle \ mathrm {GL} ^ {+} (n, \ mathbb {R})}$${\ displaystyle n ^ {2}}$${\ displaystyle \ mathrm {GL} (n, \ mathbb {R})}$

## Over finite bodies

If is a finite field with elements, where is a prime, then is a finite group of order ${\ displaystyle K}$${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle \ mathrm {GL} (n, p)}$

${\ displaystyle \ prod _ {i = 0} ^ {n-1} \ left (p ^ {n} -p ^ {i} \ right) = \ left (p ^ {n} -1 \ right) \ cdot \ left (p ^ {n} -p \ right) \ cdot \ left (p ^ {n} -p ^ {2} \ right) \ cdots \ left (p ^ {n} -p ^ {n-1} \ right).}$

This value can be determined, for example, by counting the possibilities for the matrix columns: For the first column there are assignment options (all except the zero column), for the second column there are options (all except the multiples of the first column) etc. ${\ displaystyle p ^ {n} -1}$${\ displaystyle p ^ {n} -p}$

If is a finite field with elements, where is a prime, then is a finite group of order ${\ displaystyle K}$${\ displaystyle q = p ^ {k}}$${\ displaystyle p}$${\ displaystyle \ mathrm {GL} (n, q)}$

${\ displaystyle \ prod _ {i = 0} ^ {n-1} \ left (q ^ {n} -q ^ {i} \ right) = \ left (q ^ {n} -1 \ right) \ cdot \ left (q ^ {n} -q \ right) \ cdot \ left (q ^ {n} -q ^ {2} \ right) \ cdots \ left (q ^ {n} -q ^ {n-1} \ right).}$

Note: Above the ring with elements, where is prime, the group is a finite group of order ${\ displaystyle \ mathbb {Z} _ {p ^ {k}}}$${\ displaystyle p ^ {k}}$${\ displaystyle p}$${\ displaystyle \ mathrm {GL} (n, \ mathbb {Z} _ {p ^ {k}})}$

${\ displaystyle p ^ {(k-1) n ^ {2}} \ prod _ {i = 0} ^ {n-1} \ left (p ^ {n} -p ^ {i} \ right).}$

There are some peculiarities for the general linear group over the body with 2 elements. First of all they coincide with the projective and special projective groups , that is

${\ displaystyle \ mathrm {GL} (n, 2) = \ mathrm {PGL} (n, 2) = \ mathrm {PSL} (n, 2)}$.

In particular, these groups are for simple and in small dimensions exist the following isomorphisms: ${\ displaystyle n \ geq 3}$

${\ displaystyle \ mathrm {GL} (2.2) \ cong S_ {3}}$, that is the symmetrical group S 3 with 6 elements
${\ displaystyle \ mathrm {GL} (3.2) \ cong \ mathrm {PSL} _ {2} (7)}$, that is the simple group with 168 elements
${\ displaystyle \ mathrm {GL} (4,2) \ cong A_ {8}}$, that is the alternating group A 8 with 20160 elements.

## Projective linear group

The projective linear group over a vector space over a field is the factor group , with the normal (even central ) subgroup of the scalar multiples of identity being with off . The names etc. correspond to those of the general linear group. If a finite body are and equally powerful but generally not isomorphic. ${\ displaystyle \ mathrm {PGL} (V)}$${\ displaystyle V}$${\ displaystyle K}$ ${\ displaystyle \ mathrm {GL} (V) / K ^ {\ times}}$${\ displaystyle K ^ {\ times}}$${\ displaystyle k \ cdot \ mathrm {id} _ {V}}$ ${\ displaystyle \ mathrm {id}: V \ rightarrow V}$${\ displaystyle k}$${\ displaystyle K \ setminus \ {0 \}}$${\ displaystyle \ mathrm {PGL} (n, K)}$${\ displaystyle K}$${\ displaystyle \ mathrm {PGL} (n, K)}$${\ displaystyle \ mathrm {SL} (n, K)}$

The name comes from projective geometry , where the analogue to the general linear group is the projective linear group , while the group belongs to the -dimensional projective space , it is the group of all projectivities of space. ${\ displaystyle n}$${\ displaystyle K}$${\ displaystyle \ mathrm {PGL} (n + 1, K)}$

A special case is the group of Möbius transformations , the . ${\ displaystyle \ mathrm {PGL} (2, \ mathbb {C})}$