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| PGL(2) | |
|---|---|
| Name | PGL(2) |
| Type | Projective linear group |
| Field | Various fields |
| Related | GL(2), SL(2), PSL(2) |
PGL(2)
PGL(2) denotes the projective general linear group of degree two over a field or ring and is the quotient of GL(2) by its center, giving a group of projective linear transformations on a one-dimensional projective space. It appears across algebraic number theory, algebraic geometry, hyperbolic geometry, and Galois theory and connects to classical objects such as modular groups, Möbius transformations, and automorphism groups of projective lines. The group often acts as the full automorphism group of the projective line and plays a central role in the study of conic sections, Riemann sphere automorphisms, and finite simple group classifications.
PGL(2) is defined as GL(2, K)/Z, where GL(2, K), the invertible 2×2 matrices over a field K, is quotiented by its center Z consisting of scalar matrices; this links to determinant properties and central extensions such as Schur multipliers and central simple algebras. The group inherits a topology or algebraic structure when K is a topological field such as R or C and figures in categorical descriptions like algebraic groups and group schemes; it is a linear algebraic group of dimension three related to P^1 automorphisms. Key invariants include conjugacy classes tied to eigenvalues studied in Jordan form and trace relations that reflect connections to Lie algebras such as sl(2).
PGL(2) fits into exact sequences with GL(2), SL(2), and PSL(2), where GL(2) → PGL(2) factors through SL(2) when considering determinants and quadratic forms; central quotients produce relationships with spin groups and pin groups in low dimensions. The kernel of the projection GL(2) → PGL(2) is the multiplicative group of the base field, tying PGL(2) to G_m and norm maps in field extensions and Galois cohomology descriptions. When the base field has characteristic two or nontrivial center, the comparison with PSL(2) depends on whether -I equals I, connecting to phenomena in group cohomology and covering spaces in geometric realizations.
Elements of PGL(2) are equivalence classes of 2×2 matrices; representatives act on one-dimensional projective coordinates [x:y] by linear transformations, yielding classical Möbius maps expressible as fractional linear maps familiar from Riemann sphere theory and complex analysis. This projective action identifies PGL(2) with the automorphism group of P^1 and with groups of fractional transformations appearing in Schwarzian derivative contexts and in the uniformization of Riemann surfaces such as Riemann sphere and upper half-plane models. Conjugacy classes correspond to matrix types—elliptic, parabolic, hyperbolic—terminology shared with Kleinian groups and Fuchsian groups.
Over different base fields K, PGL(2, K) exhibits distinct structural features: as an algebraic group it is isomorphic to the automorphism group of a conic when K is perfect, linking to Severi–Brauer varieties and Brauer group obstructions. Over algebraically closed fields such as C, classification reduces to conjugacy by diagonalization and Jordan form linked to Lie groups and representation theory of sl(2). Over local fields like Q_p one studies smooth representations and Bruhat–Tits buildings connected to Bruhat decomposition and Iwahori subgroups. Over finite fields F_q, structure relates to finite simple groups and central extensions in the classification of Chevalley groups.
PGL(2) has rich representation theory: continuous unitary representations when K = R occur in contexts of harmonic analysis and automorphic forms on quotients by arithmetic subgroups like SL(2,Z), linking to Maass forms and Langlands program conjectures. Algebraic representations connect to actions on projective varieties, moduli spaces such as M_g for g=0, and geometric invariant theory related to GIT quotients and stability conditions studied by David Mumford and collaborators. Dynamics of PGL(2) actions appear in Teichmüller theory, ergodic theory, and in the study of limit sets in Kleinian groups and Schottky groups.
Arithmetic applications include the role of PGL(2) in Galois representations, modularity lifting theorems of Andrew Wiles and collaborations, and in the study of rational points on conics via Hasse principles and local–global principles. Algebraic applications connect to projective models of curves, descent theory, and to arithmetic of quaternion algebras and Hilbert modular forms where local forms of PGL(2) appear in the description of automorphic representations and in the trace formulas of Atle Selberg and James Arthur.
For K = R, PGL(2,R) is isomorphic to the group of orientation-preserving and reversing Möbius transformations of the RP^1 and relates to H^2 isometries and the Lorentz group in two dimensions. For K = C, PGL(2,C) equals the full group of Möbius transformations of the Riemann sphere and is central in complex dynamics and conformal mapping. For finite fields F_q, PGL(2,F_q) yields finite simple groups for q ≥ 4 with exceptions tied to A_5 occurrences, and it appears in the classification of simple groups and in permutation representations linked to classical groups and linear fractional transformations on the projective line over F_q.
Category:Linear algebraic groups