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  "body": "\n## Definition\n\nLet $F$ be a [[field|04C5ZBG1WT]] and let $V$ be a [[finite-dimensional vector space|018W3V3TCT]] over $F$.\nA *classical group* over $F$ is one of the following groups:\n\n1. the [[general linear group|0FCHE902D7]] $\\mathrm{GL}(V)$ of invertible linear maps $V \\to V$, or the [[special linear group|0V9PMN0RBH]] $\\mathrm{SL}(V)$ of those of determinant $1$;\n2. the [[orthogonal group|0TKYAEJ8N7]] $\\mathrm{O}(V, B) = \\set{g \\in \\mathrm{GL}(V) : B(gu, gv) = B(u, v) \\text{ for all } u, v}$ of a nondegenerate symmetric [[bilinear form|0SQZMXB6GP]] $B$ on $V$, where $F$ has characteristic not $2$;\n3. the symplectic group $\\mathrm{Sp}(V, \\omega)$ of all $g \\in \\mathrm{GL}(V)$ preserving a nondegenerate alternating bilinear form $\\omega$ on $V$;\n4. the [[unitary group|0R9R0Y1HDK]] $\\mathrm{U}(V, h)$ of all $g \\in \\mathrm{GL}(V)$ preserving $h$. Here $h$ is a nondegenerate Hermitian [[sesquilinear form|054YEQ0Y8M]] for an automorphism $\\sigma$ of $F$ of order $2$;\n\ntogether with their subgroups of elements of determinant $1$, such as $\\mathrm{SO}(V, B)$ and $\\mathrm{SU}(V, h)$.\n\n## Remarks\n\nThe term is a name for this list rather than a single axiom, and authors vary.\nMany include the analogous groups over the quaternions or over other division rings, and the projective quotients such as $\\mathrm{PSL}(V)$, which give finite simple groups when $F$ is finite.\n\nExample: $\\mathrm{SO}(3)$, the special orthogonal group of the standard form $B(u, v) = u_1 v_1 + u_2 v_2 + u_3 v_3$ on $\\R^3$, is classical; $B$ is symmetric and nondegenerate, since its matrix is the identity.\n\nNon-example: the group of $g \\in \\mathrm{GL}_2(\\R)$ preserving $B(u, v) = u_1 v_1$ is not given by clause 2, since $B$ is degenerate: $B(u, (0, 1)) = 0$ for all $u$.\nThis group consists of the matrices $\\begin{pmatrix} \\pm 1 \u0026 0 \\\\ c \u0026 d \\end{pmatrix}$ with $d \\neq 0$.\n\nWhen $F = \\C$ and $\\sigma$ is complex conjugation, clause 4 gives $\\mathrm{U}(n)$ for the standard positive definite form; a nondegenerate alternating form exists only when $\\dim V$ is even.\n",
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