Projective linear group

From Wikipedia, the free encyclopedia

Relation between the projective special linear group PSL and the projective general linear group PGL.

In mathematics, especially in area of algebra called group theory, the projective linear group (also known as the projective general linear group) is one of the fundamental groups of study, part of the so-called classical groups. The projective linear group of a vector space V over a field F is the quotient group

PGL(V) = GL(V)/Z(V)

where GL(V) is the general linear group on V and Z(V) is the subgroup of all nonzero scalar transformations of V.

The projective special linear group is defined analogously:

PSL(V) = SL(V)/SZ(V)

where SL(V) is the special linear group over V and SZ(V) is the subgroup of scalar transformations with unit determinant.

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.

Note that PGL(n, F) and PSL(n, F) are equal if and only if every element of F contains a nth root in F. As an example, note that PGL(2,C)=PSL(2,C), but PGL(2,R)>PSL(2,R);^[1] this corresponds to the projective line being orientable, and the projective special linear group only being the orientation-preserving transforms.

[edit] Name

The name comes from projective geometry, where the projective group acting on homogeneous coordinates (x₀:x₁: … :x_n) 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).

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.

[edit] Finite fields

The projective special linear groups PSL(n,F_q) for a finite field F_q are often written as PSL(n,q) or L_n(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.

The special linear groups SL(n,q) are thus quasisimple: perfect central extensions of a simple group (unless n = 2 and q = 2 or 3).

[edit] Covering groups

Over the real and complex numbers, the projective special linear groups are the minimal Lie group realizations for the special linear Lie algebra every connected Lie group whose Lie algebra is is a cover of PSL(n,F). Conversely, its universal covering group is the maximal element, and the intermediary realizations form a lattice of covering groups.

For example SL₂(R) has center {±1} and fundamental group Z, and thus has universal cover and covers the centerless PSL₂(R).

[edit] Exceptional isomorphisms

In addition to the isomorphisms

and ,

there are other exceptional isomorphisms between projective special linear groups and alternating groups:

This does not make these latter projective linear groups solvable: the alternating groups over 5 or more letters are simple.

The associated extensions are universal perfect central extensions for A₄,A₅, by uniqueness of the universal perfect central extension; for , the associated extension is a perfect central extension, but not universal: there is a 3-fold covering group.

[edit] Examples

• Projective orthogonal group
• Projective unitary group
• Projective special orthogonal group
• Projective special unitary group

• Möbius group, PGL(2,C) = PSL(2,C)
• PSL(2,7)
• PSL(2,R)

[edit] See also

• Projective transformation
• Unit

[edit] References

1. ^ Gareth A. Jones and David Silverman. (1987) Complex functions: an algebraic and geometric viewpoint. Cambridge UP. Discussion of PSL and PGL on page 20 in google books

This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (February 2008)

• Grove, Larry C. (2002), Classical groups and geometric algebra, Graduate Studies in Mathematics, 39, Providence, R.I.: American Mathematical Society, MR1859189, ISBN 978-0-8218-2019-3