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

→ Main article : 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 . ${\displaystyle \mathrm {PGL} (2,\mathbb {C} )}$

Web links

• General linear group in the Encyclopaedia of Mathematics
• Eric W. Weisstein : General Linear Group . In: MathWorld (English).

Individual evidence

1. ↑ Jeffrey Overbey, William Traves and Jerzy Wojdylo: On The Keyspace Of The Hill Cipher . (PDF).