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| PSU(n,q) | |
|---|---|
| Name | Projective special unitary group |
| Notation | PSU(n,q) |
| Type | Finite simple group (for most n,q) |
| Related | PSL(n,q), PGL(n,q), SU(n,q), U(n,q), Chevalley group |
PSU(n,q)
PSU(n,q) is the projective special unitary group defined for a positive integer n≥2 and a prime power q; it is a family of finite groups arising from unitary forms over finite fields and figures centrally in the classification of finite simple groups, the theory of Chevalley groups, and the study of classical groups such as PSL(n,q) and PSp(2n,q). These groups connect to many named objects and people in algebraic group theory and finite group theory, including results of Élie Cartan, Claude Chevalley, Issai Schur, and the Atlas of Finite Groups. PSU(n,q) provides examples of groups acting on Hermitian varieties, buildings, and polar spaces studied by Jacques Tits, H.S.M. Coxeter, and Beniamino Segre.
For n≥2 and q a prime power, PSU(n,q) is obtained from the special unitary group SU(n,q) defined over the finite field GF(q^2) with the nontrivial Galois involution x↦x^q; one forms SU(n,q) as the subgroup of SL(n,q^2) preserving a nondegenerate Hermitian form, then quotients by the center to get the projective group. Construction uses concepts developed by Émile Picard in algebraic geometry, the linear algebra over Galois fields, and techniques found in works by Hermann Weyl, Nathan Jacobson, and Armand Borel. The central quotient can be nontrivial: the center intersects the special unitary group in scalars of order dividing gcd(n,q+1), a detail treated in texts by Huppert, Suzuki, and Gorenstein.
Unitary groups over finite fields were studied in the early 20th century in the context of classical groups by W.R. Hamilton's legacy and in the structural classification pursued by Wielandt and Schur. The "projective special unitary" terminology reflects the lineage from special linear and projective groups appearing in works by Camille Jordan, Emil Artin, and later systematized by Claude Chevalley and Jacques Tits within the classification of algebraic groups. Instances of PSU groups appear in the Atlas of Finite Groups compiled under editors including John Conway and Robert Curtis, and in the resolution of the classification of finite simple groups by contributors such as Daniel Gorenstein, Richard Lyons, and Ron Solomon.
PSU(n,q) inherits many algebraic properties studied for classical groups: orders can be computed by formulae involving q and n found in sources by Steinberg and Roger Carter. For n≥3 (with specific small-q exceptions), PSU(n,q) is simple except in low-rank special cases treated by Jordan and Dickson. The groups form families in the framework of algebraic groups over finite fields and relate to Frobenius endomorphisms and Lang–Steinberg theory developed by Frobenius and Robert G. Steinberg. Their Schur multipliers and outer automorphism groups were determined in investigations involving Schreier techniques and later compendia by G. James and J. H. Conway.
Maximal subgroups of PSU(n,q) incorporate stabilizers of totally isotropic subspaces, parabolic subgroups, and various classical subgroups analogous to stabilizers in orthogonal groups and symplectic groups; these are catalogued in classification work by Aschbacher and Bray, Holt, Roney-Dougal. Subgroups include projective images of unitary, linear, and extension field subgroups related to Singer cycles and cyclic subgroups considered by Zassenhaus and Feit. Exotic local subgroups and centralizers of semisimple elements connect to the analysis of centralizers in the Atlas of Finite Groups and to specific cases studied by Walter Feit and John G. Thompson.
Representation theory of PSU(n,q) uses Deligne–Lusztig theory and Lusztig's classification of irreducible characters of finite reductive groups; foundational contributors include Pierre Deligne, George Lusztig, and Nicholas Bourbaki expositions. Modular representations over fields of characteristic p and cross-characteristic representations are treated in work by Richard Brauer, Jeffrey Alperin, and Michael Aschbacher. Character tables for many small parameter values appear in computational collections associated with ATLAS projects involving Conway and J. H. Conway's collaborators and in databases used by researchers like Graham Higman and Robert Griess.
PSU(n,q) acts naturally on Hermitian varieties, polar spaces, and associated projective geometries studied by E. H. Moore, Beniamino Segre, and J. W. P. Hirschfeld; these actions yield doubly transitive and rank-three permutation representations linked to classical designs and strongly regular graphs examined by Peter J. Cameron, D. Higman, and Erdős. Buildings and BN-pairs for unitary groups were developed by Jacques Tits and provide geometric realization of subgroup structures, while connections to finite geometries influence coding theory where authors such as John H. Conway and N.J.A. Sloane applied such groups.
Exceptional isomorphisms relate small-rank PSU groups to other classical groups: for instance, low-dimensional coincidences connect certain PSU(2,q) and PSU(3,2) instances to PSL(2,q) or to sporadic groups catalogued in the Atlas of Finite Groups; these coincidences were explored by Dickson, Frobenius, and later by Sims and Curtis. Classification results situate PSU(n,q) families within the Classification of finite simple groups landscape and note sporadic exceptions and covering groups treated in monographs by Gorenstein, Lyons, and Solomon.
Category:Finite groups