Projective linear group

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Classical groups
General linear group
Special linear group
Orthogonal group
Special orthogonal group
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This article is about a certain group in mathematics. For the Iranian football (soccer) league also referred to as PGL, see Persian Gulf Cup

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 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.

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.

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] Exceptional isomorphisms

In addition to the isomorphisms

$L_2(2) \cong S_3$ and $L_2(3) \cong A_4$ ,

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

$L_2(4) \cong L_2(5) \cong A_5$

$L_2(9) \cong A_6$

$L_4(2) \cong A_8$

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

The associated extensions $\operatorname{SL}(n,q) \to \operatorname{PSL}(n,q)$ are universal perfect central extensions for $A 4, A 5$ , by uniqueness of the universal perfect central extension; for $L_2(9) \cong A_6$ , the associated extension is a perfect central extension, but not universal: there is a 3-fold covering group.

[edit] See also

[edit] References

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, vol. 39, Graduate Studies in Mathematics, Providence, R.I.: American Mathematical Society, MR 1859189, ISBN 978-0-8218-2019-3