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Group theory and generalizations Definition

Classical group

Builds on FieldFinite-dimensional vector spaceGeneral linear groupSpecial linear groupBilinear formSesquilinear formOrthogonal groupUnitary group

Definition

Let FF be a field and let VV be a finite-dimensional vector space over FF. A classical group over FF is one of the following groups:

  1. the general linear group GL(V)\mathrm{GL}(V) of invertible linear maps V→VV \to V, or the special linear group SL(V)\mathrm{SL}(V) of those of determinant 11;
  2. the orthogonal group O(V,B)={g∈GL(V):B(gu,gv)=B(u,v) for all u,v}\mathrm{O}(V, B) = \set{g \in \mathrm{GL}(V) : B(gu, gv) = B(u, v) \text{ for all } u, v} of a nondegenerate symmetric bilinear form BB on VV, where FF has characteristic not 22;
  3. the symplectic group Sp(V,ω)\mathrm{Sp}(V, \omega) of all g∈GL(V)g \in \mathrm{GL}(V) preserving a nondegenerate alternating bilinear form ω\omega on VV;
  4. the unitary group U(V,h)\mathrm{U}(V, h) of all g∈GL(V)g \in \mathrm{GL}(V) preserving hh. Here hh is a nondegenerate Hermitian sesquilinear form for an automorphism σ\sigma of FF of order 22;

together with their subgroups of elements of determinant 11, such as SO(V,B)\mathrm{SO}(V, B) and SU(V,h)\mathrm{SU}(V, h).

Remarks

The term is a name for this list rather than a single axiom, and authors vary. Many include the analogous groups over the quaternions or over other division rings, and the projective quotients such as PSL(V)\mathrm{PSL}(V), which give finite simple groups when FF is finite.

Example: SO(3)\mathrm{SO}(3), the special orthogonal group of the standard form B(u,v)=u1v1+u2v2+u3v3B(u, v) = u_1 v_1 + u_2 v_2 + u_3 v_3 on ℝ3\R^3, is classical; BB is symmetric and nondegenerate, since its matrix is the identity.

Non-example: the group of g∈GL2(ℝ)g \in \mathrm{GL}_2(\R) preserving B(u,v)=u1v1B(u, v) = u_1 v_1 is not given by clause 2, since BB is degenerate: B(u,(0,1))=0B(u, (0, 1)) = 0 for all uu. This group consists of the matrices (±10cd)\begin{pmatrix} \pm 1 & 0 \\ c & d \end{pmatrix} with d≠0d \neq 0.

When F=ℂF = \C and σ\sigma is complex conjugation, clause 4 gives U(n)\mathrm{U}(n) for the standard positive definite form; a nondegenerate alternating form exists only when dim⁡V\dim V is even.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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Bilinear form, concept, no evidence levelBilinear formconceptField, concept, no evidence levelFieldconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionSesquilinear form, concept, no evidence levelSesquilinear formconceptVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionLinear span, definition, no evidence levelLinear spandefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionSpanning set, definition, no evidence levelSpanning setdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionUnitary group, object, no evidence levelUnitary groupobjectDeterminant, definition, no evidence levelDeterminantdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionSpecial linear group, object, no evidence levelSpecial linear groupobjectClassical group, definition, no evidence levelClassical groupdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Bilinear form concept
  2. Field concept
  3. General linear group object
  4. Identity matrix object
  5. Matrix definition
  6. Sesquilinear form concept
  7. Vector space definition
  8. Square matrix definition
  9. Transpose definition
  10. Linear combination definition
  11. Linear map definition
  12. Linear subspace definition
  13. Conjugate transpose definition
  14. Orthogonal matrix definition
  15. Linear span definition
  16. Multilinear map definition
  17. Orthogonal group object
  18. Unitary matrix definition
  19. Spanning set definition
  20. Alternating multilinear map definition
  21. Unitary group object
  22. Determinant definition
  23. Finite-dimensional vector space definition
  24. Special linear group object
  25. Classical group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Classical_group
About this page
TagPQB48WPK5C
URIm2://PQB48WPK5C
Kinddefinition
WritingW0 stub
MSC20G15, 20H20
DefinesClassical group
UsesField, Finite-dimensional vector space, General linear group, Special linear group, Bilinear form, Sesquilinear form, Orthogonal group, Unitary group
Changed2 November 2026

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