|
In mathematics, the projective linear group (also known as the projective general linear group) of a vector space V over a field F is the quotient group Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...
- PGL(V) = GL(V)/Z(V)
where GL(V) is the general linear group on V and Z(V) is the group of all nonzero scalar transformations of V. 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. ...
The projective special linear group is defined analogously:
- PSL(V) = SL(V)/SZ(V)
where SZ(V) is the group of scalar transformations with unit determinant. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
Note that the groups Z(V) and SZ(V) are the centers of GL(V) and SL(V) respectively. If V is an n-dimensional vector space over a field F the alternate notations PGL(n, F) and PSL(n, F) are also used. In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
The name comes from projective geometry, where the projective group acting on homogeneous coordinates (x0:x1: … :xn) is the underlying group of the geometry (N.B. this is therefore PGL(n + 1, F) for projective space of dimension n). Stated differently, the natural action of GL(V) on V descends to an action of PGL(V) on the projective space P(V). Projective geometry is a non-metrical form of geometry. ...
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. ...
This article does not cite its references or sources. ...
In mathematics, a symmetry group describes all symmetries of objects. ...
The projective linear groups therefore generalise the case PGL(2,C) of Möbius transformations (sometimes called the Möbius group), which acts on the projective line. In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...
In geometry, a Möbius transformation is a function: where z, a, b, c, d are complex numbers satisfying ad − bc ≠0. ...
In mathematics, a projective line is a one-dimensional projective space. ...
The projective special linear groups PSL(n,Fq) for a finite field Fq are often written as PSL(n,q) or Ln(q). They are finite simple groups whenever n is at least 2, with two exceptions: L2(2), which is isomorphic to S3, the symmetric group on 3 letters, and is solvable; and L2(3), which is isomorphic to A4, the alternating group on 4 letters, and is also solvable. In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...
For a complete list see list of finite simple groups. ...
In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...
In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ...
In mathematics an alternating group is the group of even permutations of a finite set. ...
Some larger projective special linear groups are also isomorphic to alternating groups: L2(4) and L2(5) are isomorphic to A5, L2(9) is isomorphic to A6, and L4(2) is isomorphic to A8. This does not make them solvable groups as well: the alternating groups over 5 or more letters are simple.
See also
|
Feeds |
Contact