List of Projective Geometric Groups
Introduction
This article lists the projective geometric groups of the corpus — $\operatorname{PGL}$, $\operatorname{PSL}$, $\operatorname{PO}$, $\operatorname{PSO}$, $\operatorname{PU}$, $\operatorname{PSU}$ and $\operatorname{PSp}$ — the quotients of the linear and classical groups by their centres, each with the form or the structure it preserves and the article that defines it. The passage to the projective group is the descent from a vector space to its space of lines: the scalars act trivially on the lines, and the quotient by the centre is exactly the group of transformations the projective geometry sees. The projective space itself appears in the list as a homogeneous space, $\mathbb{P}(V) = \operatorname{GL}(V)/P$ with $P$ the stabiliser of a line.
Every entry points to the article that introduces the group or the homogeneous space. The article introduces nothing and proves nothing: it records the quotient, the centre, the simplicity statement and the preserving structure as the introducing articles give them, and it neither restates a definition nor gives a proof.
The article records examples and non-examples side by side. Beside the projective groups it lists the cases in which the quotient behaves unexpectedly: $\operatorname{PGL}_n$ differs from $\operatorname{PSL}_n$ when $F^\times$ has non-trivial $n$-th power classes; the projective groups of a general field include the semilinear collineations $\operatorname{P\Gamma L}$ not generated by the linear ones; $\operatorname{PSL}_2(2) \cong S_3$, $\operatorname{PSL}_2(3) \cong A_4$ and $\operatorname{PSU}_3(2)$ are not simple; and the spin groups, which cover the projective orthogonal groups but are not themselves quotients by the centre, each with the failure named and the article that records it.
The Projective Quotients of the Linear Groups
The projective general and projective special linear groups are the quotients of $\operatorname{GL}(V)$ and $\operatorname{SL}(V)$ by their centres, and over a field they are the collineation groups of the projective space.
| Group | The property it has | Introduced in |
|---|---|---|
| $\operatorname{PGL}(V) = \operatorname{GL}(V)/F^\times$ | the quotient by the scalars; the group of linear collineations of $\mathbb{P}(V)$ | The General Linear Group |
| $\operatorname{PSL}(V) = \operatorname{SL}(V)/\mu_n$ | the quotient of $\operatorname{SL}(V)$ by its centre $\mu_n = \operatorname{SL}(V) \cap F^\times\mathrm{id}$ | The General Linear Group; The Special Linear Group and the Determinant |
| the quotient $\operatorname{PGL}_n/\operatorname{PSL}_n$ | isomorphic to $F^\times/(F^\times)^n$; trivial exactly when every element of $F^\times$ is an $n$-th power | The General Linear Group |
| the fundamental theorem of projective geometry | a collineation of $\mathbb{P}(V)$, $\dim V \geq 3$, comes from a semilinear map; over a field it lies in $\operatorname{PGL}(V)$ | Projective Geometry |
| the semilinear collineation group $\operatorname{P\Gamma L}(V)$ | the group generated by $\operatorname{PGL}(V)$ and the field automorphisms; strictly larger over a non-prime field | Projective Geometry |
| the projective line $\mathbb{P}^1$ | $\operatorname{PGL}_2$ acts sharply three-transitively; the cross ratio is its invariant | Projective Geometry |
| $\operatorname{PGL}_2(\mathbb{R})$, $\operatorname{PGL}_2(\mathbb{C})$ | the Möbius groups of the circle and of the Riemann sphere | Möbius and Lie Sphere Geometry |
| $\operatorname{PSL}_2(\mathbb{R})$, $\operatorname{PSL}_2(\mathbb{C})$ | the orientation-preserving isometries of the hyperbolic plane and of hyperbolic three-space | Möbius and Lie Sphere Geometry; List of Möbius and Conformal Groups |
The Projective Quotients of the Classical Groups
Each classical group has a centre, generated by $\pm I$ or by the scalars, and the quotient is the projective group of the corresponding form; the projective group is what acts on the projective quadric.
| Group | The structure it preserves | Introduced in |
|---|---|---|
| $\operatorname{PO}(V,q) = \operatorname{O}(V,q)/\{\pm I\}$ | the projective quadric $Q = \{[v] : q(v) = 0\}$ and its polarity | Projective Geometry; Matrix Groups and Classical Groups |
| $\operatorname{PSO}(V,q) = \operatorname{SO}(V,q)/Z$ | the quadric with its orientation; $Z = \{\pm I\}$ for even $n$ and trivial for odd $n$ | The Rotation Group and Orientation; Projective Geometry |
| $\operatorname{PU}(V,h) = \operatorname{U}(V,h)/Z$ | the projective Hermitian form and its unitary polarity | The Unitary and Symplectic Groups |
| $\operatorname{PSU}(V,h) = \operatorname{SU}(V,h)/Z$ | the determinant-one projective unitary group; $Z = Z(SU(n)) \cong \mathbb{Z}/n\mathbb{Z}$ | The Unitary and Symplectic Groups; Matrix Groups and Classical Groups |
| $\operatorname{PSp}(V,\omega) = \operatorname{Sp}(V,\omega)/\{\pm I\}$ | the projective symplectic structure and the projective Lagrangians | The Unitary and Symplectic Groups; Symplectic Forms and Poisson Brackets |
| $PO(6) \cong PGL(4)$ | the Klein quadric of the lines of $\mathbb{P}^3$; the projective form of $\operatorname{Spin}(6) \cong SL(4,\mathbb{C})$ | Projective Geometry |
| the central isogeny of a form | the sequence $1 \to Z \to G \to PG \to 1$; the half-spin representations of the orthogonal case | Matrix Groups and Classical Groups; List of Clifford Algebras and Spin Groups |
The Projective Space and the Grassmannians as Homogeneous Spaces
The projective space and the Grassmannians are the homogeneous spaces of the linear groups, and the projective quadric is the homogeneous space of the projective orthogonal group.
| Homogeneous space | The description | Introduced in |
|---|---|---|
| $\mathbb{P}(V) = \operatorname{GL}(V)/P$ | $P$ the stabiliser of a line, a parabolic subgroup; $\mathbb{P}(V)$ has dimension $n-1$ | The General Linear Group; Homogeneous Spaces |
| $\mathbb{P}(V) = \operatorname{PGL}(V)/H$ | the same space as a homogeneous space of the projective group; $H$ the stabiliser of a point | The General Linear Group; Homogeneous Spaces |
| the Grassmannian $\operatorname{Gr}_k(V)$ | $\operatorname{GL}(V)/P_k$ with $P_k$ the stabiliser of a $k$-subspace | The General Linear Group |
| the flag variety | $\operatorname{GL}(V)/B$ with $B$ the Borel subgroup of upper triangular matrices | The General Linear Group |
| the projective quadric $Q$ | $\operatorname{PO}(V,q)/H$ with $H$ the stabiliser of a point; a symmetric space in the real definite case | Projective Geometry; Homogeneous Spaces |
| the Klein quadric $Q_K$ | the lines of $\mathbb{P}^3$ as the points of a quadric in $\mathbb{P}^5$; the Plücker embedding | Projective Geometry |
| the Gaussian binomial $\binom{n}{k}_q$ | the number of $k$-subspaces of $\mathbb{F}_q^n$, the order of the Grassmannian | Projective Geometry |
Simplicity and the Exceptions
The projective classical groups over a finite field or over a field are simple except for a finite list of small cases, and the simplicity is the source of the classical simple groups.
| Statement | The content | Introduced in |
|---|---|---|
| the simplicity of the projective classical groups | $\operatorname{PSL}_n(q)$, $\operatorname{PSp}_{2n}(q)$, $\operatorname{PSU}_n(q)$ and $\mathrm{P}\Omega^\pm_{2n}(q)$ are simple for the parameters where the root system is simple and the group is the adjoint or simply connected group modulo its centre | Finite Simple Groups of Lie Type |
| the exceptions | $\operatorname{PSL}_2(2) \cong S_3$, $\operatorname{PSL}_2(3) \cong A_4$, $\operatorname{PSU}_3(2)$ solvable of order $72$, $\operatorname{Sp}_4(2) \cong S_6$ with simple derived subgroup $A_6$ | Finite Simple Groups of Lie Type |
| the coincidences | $\operatorname{PSL}_2(4) \cong \operatorname{PSL}_2(5) \cong A_5$, $\operatorname{PSL}_2(7) \cong \operatorname{PSL}_3(2)$, $\operatorname{PSL}_2(9) \cong A_6$, $\operatorname{PSL}_4(2) \cong A_8$, $\operatorname{PSp}_4(3) \cong \operatorname{PSU}_4(2)$ | Finite Simple Groups of Lie Type |
| $\operatorname{PSL}_2(q)$ | simple for $q \geq 4$; the doubly transitive action on the projective line is the classical proof | Finite Simple Groups of Lie Type |
| the orders | $\lvert\operatorname{PSL}_n(q)\rvert = \lvert\operatorname{SL}_n(q)\rvert/\gcd(n,q-1)$, with the analogous divisions by the centre in the other families | Finite Simple Groups of Lie Type |
| the place in the classification | the finite simple groups of Lie type are the adjoint groups of the simple root systems and their twisted forms; the projective classical groups are the adjoint forms of $A$, $B$, $C$, $D$ | Finite Simple Groups of Lie Type; The Classification of Finite Simple Groups |
The Projective Groups over the Classical Fields
The base field decides which quotient occurs and what the group is: over an algebraically closed field the two linear quotients coincide, over $\mathbb{R}$ and $\mathbb{Q}$ they differ, and the low-dimensional projective groups are the classical groups of the Möbius geometry.
| Object or identification | The statement | Introduced in |
|---|---|---|
| $A_n = \mathrm{PSL}_{n+1}$ | the classical family $A_n$ is the projective special linear group; $\operatorname{PGL}_n$ is the adjoint form | Finite Simple Groups of Lie Type; Root Systems and Classification |
| $\operatorname{PGL}_n(F) = \operatorname{PSL}_n(F)$ over an algebraically closed field | every element has an $n$-th root, so the $n$-th power classes are trivial | The General Linear Group |
| $\operatorname{PGL}_n(\mathbb{R})$ and $\operatorname{PSL}_n(\mathbb{R})$ | the real projective groups, of dimension $n^2-1$; they differ when $n$ is even | The General Linear Group; Lie Groups |
| the projective space $\mathbb{RP}^{n-1}$, $\mathbb{CP}^{n-1}$ | the homogeneous spaces of the real and complex projective groups | Projective Geometry; Homogeneous Spaces |
| $\operatorname{PGL}(2,\mathbb{R}) \cong PO(2,1)$ | the projective linear group of the line is the projective orthogonal group of the split conic; the full isometry group of $\mathbb{H}^2$ | List of Möbius and Conformal Groups; Projective Geometry |
| $\operatorname{PSL}(2,\mathbb{R}) \cong SO^+(2,1)$ | the real Möbius group is the identity component of the orthogonal group of signature $(2,1)$ | List of Möbius and Conformal Groups; Hyperbolic Geometry |
| $\operatorname{PSL}(2,\mathbb{C}) \cong SO^+(1,3)$ | the complex Möbius group is the proper orthochronous Lorentz group | List of Möbius and Conformal Groups |
| $\operatorname{PGL}_n(\mathbb{F}_q)$ and $\operatorname{PSL}_n(\mathbb{F}_q)$ | the finite projective groups; $\lvert\operatorname{PSL}_n(q)\rvert = \lvert\operatorname{SL}_n(q)\rvert/\gcd(n,q-1)$ | Finite Simple Groups of Lie Type |
| the adjoint group $G_{\mathrm{ad}}$ | the quotient of the simply connected group by its centre; $\operatorname{PGL}_n$ is the adjoint form of $A_{n-1}$ | Finite Simple Groups of Lie Type |
| the automorphism group of the projective space | $\operatorname{Aut}(\mathbb{P}(V)) \cong \operatorname{PGL}(V) \rtimes \operatorname{Aut}(F)$ when every collineation is induced, for $\dim V \geq 3$ | The General Linear Group |
| the plane case | the fundamental theorem fails for $\dim V = 2$; the projective line is too small and the low-dimensional groups behave exceptionally | The General Linear Group; Projective Geometry |
Non-examples and Warnings
| Object | Why the expected statement fails | Introduced in |
|---|---|---|
| $\operatorname{PGL}_n(F)$ when $(F^\times)^n \neq F^\times$ | it is strictly larger than $\operatorname{PSL}_n(F)$; the quotient is $F^\times/(F^\times)^n$ | The General Linear Group |
| the collineation group over a non-prime field | $\operatorname{P\Gamma L}(V)$ contains the semilinear maps and is larger than $\operatorname{PGL}(V)$; the fundamental theorem gives only the linear part | Projective Geometry |
| $\operatorname{PSL}_2(2) \cong S_3$ | not simple; the smallest member of the family is $S_3$ and $\operatorname{PSL}_2(2) = \operatorname{PGL}_2(2)$ | Finite Simple Groups of Lie Type |
| $\operatorname{PSU}_3(2)$ | solvable of order $72$; an exception to the simplicity of the unitary family | Finite Simple Groups of Lie Type |
| the centre of $\operatorname{O}(V,q)$ | it is $\{\pm I\}$, so $\operatorname{PO}$ quotients by a group of order two; the centre is not a group of scalars as in $\operatorname{PGL}$ | Matrix Groups and Classical Groups |
| the orthogonal group of a definite real form | $\operatorname{O}(n)$ is not simple and not connected; $\operatorname{PO}(n)$ is the simple projective image, not $\operatorname{O}(n)$ itself | Matrix Groups and Classical Groups |
| the class of $\operatorname{PO}$ in the $\mathbb{F}_q$ simplicity list | the group $\mathrm{P}\Omega$ of the classification is the adjoint form, a quotient of $\operatorname{PSO}$ by a further subgroup in the orthogonal case, so the names do not always coincide | Finite Simple Groups of Lie Type |
Objects that a reader may expect in a list of projective geometric groups, and does not find here.
| Object | Why it is not listed | Introduced in |
|---|---|---|
| $\operatorname{GL}(V)$, $\operatorname{SL}(V)$ | they are the linear groups, not their quotients | List of Linear Geometric Groups |
| $\operatorname{O}(V,q)$, $\operatorname{U}(V,h)$, $\operatorname{Sp}(V,\omega)$ | they are the classical groups themselves; only their projective images are listed here | List of Classical Geometric Groups |
| the spin and pin groups | they are the double covers of the orthogonal and projective orthogonal groups, constructed from the Clifford algebra, and are not quotients by the centre | List of Clifford Algebras and Spin Groups |
| the Möbius group $\operatorname{Möb}(n) = O(n+1,1)/\{\pm1\}$ | it is a projective orthogonal group, but it is catalogued with the Möbius and conformal groups | List of Möbius and Conformal Groups |
| the modular group $\operatorname{PSL}_2(\mathbb{Z})$ | the arithmetic subgroup is recorded with the discrete geometric groups and the lattices | List of Discrete Geometric Groups |
Summary
This article has listed the projective geometric groups of the corpus: the projective quotients $\operatorname{PGL}(V)$ and $\operatorname{PSL}(V)$ of the linear groups with the criterion for their coincidence and the semilinear group $\operatorname{P\Gamma L}$; the projective quotients $\operatorname{PO}$, $\operatorname{PSO}$, $\operatorname{PU}$, $\operatorname{PSU}$ and $\operatorname{PSp}$ of the classical groups, each with the form it preserves and the centre it divides by; the projective space and the Grassmannians as homogeneous spaces, $\mathbb{P}(V) = \operatorname{GL}(V)/P$, together with the projective quadric and the Klein correspondence $PO(6) \cong PGL(4)$; and the simplicity of the projective classical groups with its finite list of exceptions and coincidences. Beside the examples stand the non-examples: the discrepancy $\operatorname{PGL}_n \neq \operatorname{PSL}_n$, the semilinear collineations, the small non-simple members and the distinction between $\operatorname{PSO}$ and the adjoint group $\mathrm{P}\Omega$ of the classification. The list introduces and proves nothing; it is the index of the projective groups of the corpus.
Summary of Notation
A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are those of the introducing articles.
| Symbol | Meaning |
|---|---|
| $\mathbb{P}(V)$, $\mathbb{P}^n$ | the projective space of lines in $V$; the points are the lines |
| $\operatorname{GL}(V)$, $\operatorname{SL}(V)$ | the general and special linear groups |
| $\mu_n$ | the $n$-th roots of unity, the centre of $\operatorname{SL}_n(F)$ |
| $\operatorname{PGL}(V) = \operatorname{GL}(V)/F^\times$ | the projective general linear group |
| $\operatorname{PSL}(V) = \operatorname{SL}(V)/\mu_n$ | the projective special linear group |
| $\operatorname{P\Gamma L}(V)$ | the collineation group including the semilinear maps |
| $\operatorname{O}(V,q)$, $\operatorname{SO}(V,q)$, $\operatorname{PO}(V,q)$, $\operatorname{PSO}(V,q)$ | the orthogonal groups and their projective quotients |
| $\operatorname{U}(V,h)$, $\operatorname{SU}(V,h)$, $\operatorname{PU}(V,h)$, $\operatorname{PSU}(V,h)$ | the unitary groups and their projective quotients |
| $\operatorname{Sp}(V,\omega)$, $\operatorname{PSp}(V,\omega)$ | the symplectic group and its projective quotient |
| $\mathrm{P}\Omega^\pm_{2n}(q)$ | the adjoint orthogonal groups of the finite classification |
| $P$, $P_k$, $B$ | the parabolic stabilisers of a line and a $k$-subspace, the Borel subgroup |
| $\operatorname{Gr}_k(V)$, $\binom{n}{k}_q$ | the Grassmannian, the Gaussian binomial counting its points |
| $Q$, $Q_K$ | the projective quadric, the Klein quadric |
Further Reading
- Larry C. Grove, Classical Groups and Geometric Algebra (American Mathematical Society, 2002), for the projective quotients of the linear and classical groups and the collineation groups.
- Jean A. Dieudonné, La géométrie des groupes classiques (Springer, 3rd ed. 1971), for the projective classical groups over a general field and the fundamental theorem.
- John D. Dixon and Brian Mortimer, Permutation Groups (Springer, 1996), for the simplicity of the projective classical groups and the exceptional isomorphisms.
- Roger Howe, Topics in Classical Invariant Theory (Yale University Press, 1989), for the homogeneous spaces and the projective geometry of the classical groups.