Group general linear

In Mathematical, the linear general group of degree N of a body E is the group matrices N × invertible N with coefficients in E , provided with the matric multiplication. It is noted GL_n ( E ), or GL_n (here GL ( N , E )). These groups are important in the theory of the representations of groups and appear at the time of the study of the Symétrie S and the Polynôme S.

GL ( N , E ) and its sub-groups are often called “linear groups” or “matric groups”. The linear special group, noted SSL ( N , E ) and made up of the matrices of determinant 1, is a Sous-groupe of GL ( N , E ).

Description

August 1st If N ≥ 3, GL ( N , E ) is not abelian.

Group general linear

of a vector space

If U is a vector Space on the body E , one calls linear general group of U and one notes GL ( U ) or Aut ( U ), the group of the automorphisms of U provided with the composition of the functions.

If the Dimension of U is N , then GL ( U ) and GL ( N , E ) are isomorphous. This isomorphism is not canonical and depends on the choice of a bases U . Once this base chosen, any automorphism of U can be represented by a matrix invertible N × N which determines isomorphism.

On realities and the complexes

If the body E is

\ mathbb R

(the real numbers) or

\ mathbb C

(the complex numbers), then GL ( N ) is a Groupe of Dregs real or complex of dimension N ². Indeed, GL ( N ) is consisted of the matrices of determinant not-no one. The determinant being a application continuous (and even polynomial), GL ( N ) is a subset not-vacuum of the variety of the matrices N × N , of dimension N ².

The Algèbre of Dregs associated to GL ( N ) is formed by the matrices N × real or complex N .

If GL ( N , $\ mathbb C$ ) is Connexe, GL ( N , $\ mathbb R$ ) has two components related S: matrices of positive determinant and those of negative determinant. The matrices N × real N of positive determinant train a sub-group of GL ( N , $\ mathbb R$ ), noted GL⁺ ( N , $\ mathbb R$ ). This last is also a group of Dregs of dimension N ² and has the same algebra of Dregs. It is simply related.

On the finished bodies

If E is a Corps finished of Q elements, then one writes sometimes GL ( N , Q ) in the place of GL ( N , E ). GL ( N , Q ) is a group finished of ( Q ^N - 1) ( Q ^N - Q ) ( Q ^N - Q ²)… ( Q ^N - Q ^{N -1}) elements (what can be proven by counting the number of possible columns of the matrix: the first column can be any, put aside the null column, the second any, except the multiples of the first, etc)

Linear special group

The linear special Group of order N of a body E , noted SSL ( N , E ), is the group of the matrices of determinant 1. SSL ( N , E ) is a Sous-groupe distinguished from GL ( N , E ).

If one considers E ^× ( E private of its null element), then the determinant is a Homomorphisme of group:

det: GL (N, E) $\ rightarrow$ E^×

The core of this application is the linear special group. according to the first theorem of isomorphism, GL ( N , E ) /SL ( N , E ) is isomorphous with E ^×. In fact, GL ( N , E ) can be regarded as the semi-direct Produit SSL ( N , E ) by E ^×:

GL ( N , E ) = SSL ( N , E ) ⋊ E ^×

When E is $\ mathbb R$ or $\ mathbb C$ , SSL ( N ) is the sub-group of Dregs of GL ( N ) of dimension N ²-1. The algebra of Dregs of SSL ( N ) is formed of the matrices N × N with real or complex coefficients of null trace.

Special group linear SSL ( N , $\ mathbb R$ ) can be seen as the group of the linear transformations of $\ mathbb R^n$ preserving volume and the orientation. It is generated by the Transvection S.

Linear projective group

The linear projective group of a vector space U on a body E is the Groupe quotient GL ( U ) /Z ( U ). Notations PGL ( U ), PSL ( U ), etc are similar to those used for the linear general group.

This denomination comes from the projective Géométrie, where the projective group acting on the homogeneous coordinates ( X ₀: X ₁: …: X _N) is the subjacent group of this geometry (consequently, should be considered group PGL ( N +1, E ) for a projective space of dimension N ). The linear projective group thus generalizes group PGL (2) of the transformations of Möbius, sometimes called the Groupe of Möbius.

Special projective group linear PSL ( N , E _q) of a body finished E _q is sometimes noted L_n (Q) . They are simple groups finished when N is at least equal to 2, except L ₂ (2) and L ₂ (3).

Sub-groups

Diagonal

The whole of the diagonal matrices of determinant not no form a sub-group of GL ( N , E ) isomorphous with ( E ^×) ⁿ. In the bodies

\ mathbb R

and

\ mathbb C

, it is about the group of dilations and contractions.

A scalar matrix is a diagonal matrix which is the product of the matrix identity by a constant. The whole of the nonnull scalar matrices, sometimes noted Z ( N , E ), form a sub-group of GL ( N , E ) isomorphous with E ^×. This group is the center of GL ( N , E ). It is invariant and abelian.

The center of SSL ( N , E ), noted SZ ( N , E ), is simply the whole of the scalar matrices of determinant 1. It is isomorphous with the group of the nth roots of 1.

Traditional

The traditional groups are the sub-groups of GL ( U ) which preserve part of the internal product on U . For example:

the orthogonal Group, O ( U ), which preserves a symmetrical bilinear form on U
the Groupe symplectic, Sp ( U ), which preserves an antisymmetric bilinear form on U
the unit Groupe, U ( U ), which preserves a square form on U (when E is $\ mathbb C$ ).

These groups are important examples of groups of Dregs.

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