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In mathematics, the special linear group SL(n, R) of degree n over a commutative ring R is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant
Special groups include the general linear group, the special linear group, and the symplectic group. Special groups are necessarily connected. Products of special groups are special. The projective linear group is not special because there exist Azumaya algebras, which are trivial over a finite separable extension, but not over the base field.
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group.
PSL(2, 7) is a maximal subgroup of the Mathieu group M 21; the groups M 21 and M 24 can be constructed as extensions of PSL(2, 7). These extensions can be interpreted in terms of the tiling of the Klein quartic, but are not realized by geometric symmetries of the tiling.
In mathematics, the special linear group SL(2, R) or SL 2 (R) is the group of 2 × 2 real matrices with determinant one: (,) = {():,,, =}.It is a connected non-compact simple real Lie group of dimension 3 with applications in geometry, topology, representation theory, and physics.
The classical groups are exactly the general linear groups over ℝ, ℂ and ℍ together with the automorphism groups of non-degenerate forms discussed below. [5] These groups are usually additionally restricted to the subgroups whose elements have determinant 1, so that their centers are discrete. The classical groups, with the determinant 1 ...
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group. PGL(V) = GL(V) / Z(V)
This family of groups includes the special linear groups SL(n, R) for n ≥ 3 and the special orthogonal groups SO(p,q) for p > q ≥ 2 and SO(p,p) for p ≥ 3. More generally, this holds for simple algebraic groups of rank at least two over a local field. The pairs (R n ⋊ SL(n, R), R n) and (Z n ⋊ SL(n, Z), Z n) have relative property (T ...