Projective linear group

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.

The name comes from projective geometry, where the projective group acting on homogeneous coordinates ("x"₀:"x"₁: … :"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: "L"₂(2), which is isomorphic to "S"₃, the symmetric group on 3 letters, and is solvable; and "L"₂(3), which is isomorphic to "A"₄, 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).

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 $operatorname\{SL\}(n,q)\; o\; operatorname\{PSL\}(n,q)$ 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.

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)

ee also

*Unit

References

*Citation | last1=Grove | first1=Larry C. | title=Classical groups and geometric algebra | publisher=American Mathematical Society | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-2019-3 | id=MathSciNet | id = 1859189 | year=2002 | volume=39