Projective linear group

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

In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group

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

where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V; these are quotiented out because they act trivially on the projective space and are exactly the kernel, and the notation "Z" is because the scalar transformations are the center of the general linear group.

The projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly:

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. Here SZ is the center of SL, and is naturally identified with the group of nth roots of unity in K (where n is the dimension and K is the base field).

PGL and PSL are some of the fundamental groups of study, part of the so-called classical groups, and an element of PGL is called a projective linear transformation. If V is the n-dimensional vector space over a field F, namely $V=F^{n},$ 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); this corresponds to the real projective line being orientable, and the projective special linear group only being the orientation-preserving transformations.

PGL and PSL can also be defined over a ring, with the most important example being the modular group, PSL(2, Z).

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. 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.

Note that unlike the general linear group, which is generally defined axiomatically as "invertible functions preserving the linear (vector space) structure", the projective linear group is defined constructively, as a quotient of the general linear group of the associated vector space, rather than axiomatically as "invertible functions preserving the projective linear structure". This is reflected in the notation: PGL(n,F) is the group associated to GL(n,F), and is the projective linear group of $(n-1)$-dimensional projective space, not n-dimensional projective space.

Collineations

Main article: Collineation

A related group is the collineation group, which is defined axiomatically. A collineation is an invertible (or more generally one-to-one) map which sends collinear points to collinear points. One can define a projective space axiomatically in terms of an incidence structure (a set of points P, lines L, and an incidence relation I specifying which points lie on which lines) satisfying certain axioms – an automorphism of a projective space thus defined then being an automorphism f of the set of points and an automorphism g of the set of lines, preserving the incidence relation, which is exactly a collineation of a space to itself. Projective linear transforms are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transforms map planes to planes, so projective linear transforms map lines to lines), but in general not all collineations are projective linear transforms – PGL is in general a proper subgroup of the collineation group.

Specifically, for $n=2$ (a projective line), all points are collinear, so the collineation group is exactly the symmetric group of the points of the projective line, and except for $\mathbf {F} _{2}$ and $\mathbf {F} _{3}$ (where PGL is the full symmetric group), PGL is a proper subgroup of the full symmetric group on these points.

For $n\geq 3,$ the collineation group is the projective semilinear group, $P\Gamma L$ – this is PGL, twisted by field automorphisms; formally, $P\Gamma L\cong PGL\rtimes \operatorname {Gal} (K/k),$ where k is the prime field for K; this is the fundamental theorem of projective geometry. Thus for K a prime field ($\mathbf {F} _{p}$ or $\mathbf {Q}$), we have $PGL=P\Gamma L,$ but for K not a prime field (such as $\mathbf {F} _{p^{n}}$ for $n\geq 2$ or $\mathbf {R}$), the projective linear group is a proper subgroup of the collineation group, which can be thought of as "transforms preserving a projective semi-linear structure". Correspondingly, the quotient group $P\Gamma L/PGL=\operatorname {Gal} (K/k)$ corresponds to "choices of linear structure", with the identity (base point) being the existing linear structure.

One may also define collineation groups for axiomatically defined projective spaces, where there is no natural notion of a projective linear transform. However, with the exception of the non-Desarguesian planes, all projective spaces are the projectivization of a linear space over a division ring though, as noted above, there are multiple choices of linear structure, namely a torsor over $\operatorname {Gal} (K/k)$ (for $n\geq 3$).

Properties

• PGL sends collinear points to collinear points (it preserves projective lines), but it is not the full collineation group, which is instead either PΓL (for $n>2$) or the full symmetric group for $n=2$ (the projective line).
• Every algebraic automorphism of a projective space is projective linear.
• PGL acts faithfully on projective space: non-identity elements act non-trivially.

  Concretely, the kernel of the action of GL on projective space is exactly the scalar maps, which are quotiented out in PGL.

• PGL acts 2-transitively on projective space.

  This is because 2 distinct points in projective space correspond to 2 vectors that do not lie on a single linear space, and hence are linearly independent, and GL acts transitively on k-element sets of linearly independent vectors.

• PGL(2, K) acts 3-transitively on the projective line.

  3 arbitrary points are conventionally mapped to $[0,1],[1,1],[1,0];$ in alternative notation, $0,1,\infty .$ In fractional linear transformation notation, the function ${\frac {x-a}{x-c}}\cdot {\frac {b-c}{b-a}}$ maps $a\mapsto 0,b\mapsto 1,c\mapsto \infty .$ This is the cross-ratio $(x,b;a,c)$ – see cross-ratio: transformational approach for details.

• For $n\geq 3,$ PGL(n, K) does not act 3-transitively, because it must send 3 collinear points to 3 other collinear points, not an arbitrary set. For $n=2$ the space is the projective line, so all points are collinear and this is no restriction.
• PGL(2, K) does not act 4-transitively on the projective line (except for $\operatorname {PGL} (2,\mathbf {F} _{3}),$ as $\mathbf {P} _{3}^{1}$ has 3+1=4 points, so 3-transitive implies 4-transitive); the invariant that is preserved is the cross ratio. Thus it is not the full collineation group of the projective line (except for $\mathbf {F} _{2}$ and $\mathbf {F} _{3}$).

Fractional linear transformations

Further information: Möbius transformation § Projective matrix representations

As for Möbius transformations, the group PGL(2, K) can be interpreted as fractional linear transformations with coefficients in K, a matrix $\left({\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right)$ corresponding to the rational function

$f(x)={\frac {ax+b}{cx+d}}$

where multiplication of matrices agrees with composition of functions, and quotienting out by scalar matrices corresponding to multiplying the top and bottom of the fraction by a common factor. As with Möbius transformations, these functions can be interpreted as automorphisms of the projective line over K.

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).

Order

The order of PGL(n,q) is

(qⁿ − 1)(qⁿ − q)(qⁿ − q²) … (qⁿ − qⁿ⁻¹)/(q − 1)

which corresponds to the order of GL(n,q), divided by $(q-1)$ for projectivization; see q-analog for discussion of such formulas. This also equals the order of SL(n,q); there dividing by $(q-1)$ is due to the determinant.

The order of PSL(n,q) is the above, divided by $|\operatorname {SZ} (n,q)|$, the number of scalar matrices with determinant 1 – or equivalently dividing by $|F^{*}/(F^{*})^{n}|$, the number of classes of element that have no nth root, or equivalently, dividing by the number of nth roots of unity in $\mathbf {F} _{q}$.

Exceptional isomorphisms

In addition to the isomorphisms

$L_{2}(2)\cong S_{3},L_{2}(3)\cong A_{4},$, and $\operatorname {PGL} (2,3)\cong S_{4},$

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.

The groups over $\mathbf {F} _{5}$ have a number of exceptional isomorphisms:

$\operatorname {PSL} (2,5)\cong A_{5};$

$\operatorname {PGL} (2,5)\cong S_{5},$ the symmetric group on five elements;

$\operatorname {SL} (2,5)\cong 2I,$ the binary icosahedral group.

They can also be used to give a construction of an exotic map S₅ → S₆, as described below.

A further isomorphism is:

$L_{2}(7)\cong L_{3}(2)$ is the simple group of order 168, the second smallest non-abelian simple group, and is not an alternating group; see PSL(2,7).

Action on projective line

Some of the above maps can be seen directly in terms of the action of PSL and PGL on the associated projective line: $\operatorname {PGL} (n,q)$ acts on the projective space $\mathbf {P} _{q}^{n-1},$ which has $(q^{n}-1)/(q-1)$ points, and this yields a map from the projective linear group to the symmetric group on $(q^{n}-1)/(q-1)$ points. For $n=2$, this is the projective line $\mathbf {P} _{q}^{1},$ which has $(q^{2}-1)/(q-1)=q+1$ points, so there is a map $\operatorname {PGL} (2,q)\to S_{q+1}$.

To understand these maps, it is useful to recall these facts:

• The order of $\operatorname {PGL} (2,q)$ is $(q^{2}-1)(q^{2}-q)/(q-1)=q^{3}-q;$ the order of $\operatorname {PSL} (2,q)$ either equals this (if the characteristic is 2), or is half this (if the characteristic is not 2).
• The action of the projective linear group on the projective line is faithful and 3-transitive, so the map is one-to-one and has image a 3-transitive subgroup.

Thus the image is a 3-transitive subgroup of known order, which allows it to be identified. This yields the following maps:

• $\operatorname {PSL} (2,2)=\operatorname {PGL} (2,2)\to S_{3},$ of order 6, which is an isomorphism.
• $\operatorname {PSL} (2,3)<\operatorname {PGL} (2,3)\to S_{4},$ of orders 12 and 24, the latter of which is an isomorphism, with PSL(2,3) being the alternating group.
• $\operatorname {PSL} (2,4)=\operatorname {PGL} (2,4)\to S_{5},$ of order 60, yielding the alternating group $A_{5}.$
• $\operatorname {PSL} (2,5)<\operatorname {PGL} (2,5)\to S_{6},$ of orders 60 and 120, which yields an embedding of $S_{5}$ (respectively, $A_{5}$) as a transitive subgroup of $S_{6}$ (respectively, $A_{6}$). This is an example of an exotic map S₅ → S₆, and can be used to construct the exceptional outer automorphism of S₆.

Modular group

Main article: Modular group

The groups PSL(2, Z/n) arise in studying the modular group, PSL(2, Z), as quotients by reducing all elements mod n – see also principal congruence subgroup. An interesting example is $PSL(2,Z)\to PSL(2,Z/2)\cong S_{3},$ which splits via the matrices:

$x$	$1/(1-x)$	$(x-1)/x$
${\begin{pmatrix}1&0\\0&1\end{pmatrix}}$	${\begin{pmatrix}0&1\\-1&1\end{pmatrix}}$	${\begin{pmatrix}1&-1\\1&0\end{pmatrix}}$
$1/x$	$1-x$	$x/(x-1)$
${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$	${\begin{pmatrix}-1&1\\0&1\end{pmatrix}}$	${\begin{pmatrix}1&0\\1&-1\end{pmatrix}}$

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 ${\mathfrak {sl}}_{n}:$ every connected Lie group whose Lie algebra is ${\mathfrak {sl}}_{n}$ 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 ${\overline {\mathrm {SL} _{2}(\mathbf {R} )}}$ and covers the centerless PSL₂(R).

Examples

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

Subgroups

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

See also

• Projective transformation
• Unit

References

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