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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
The group GL(n, F) and its subgroups are often called linear groups or matrix groups (the automorphism group GL(V) is a linear group but not a matrix group). These groups are important in the theory of group representations , and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of ...
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 ...
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 projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane .
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 special linear group is a complex semisimple Lie group. For any such group G and maximal compact subgroup K, and V a complex vector space of finite dimension which is a G-module, its G-submodules and K-submodules are the same. [3] In the Encyclopedia of Mathematics, the formulation is
Throughout his time in the UK, he researched linear groups. During this period, his work was influenced by David Mumford , who lectured at the University of Warwick during Lehrer's time there. [ 1 ] Upon returning to Australia, Lehrer collaborated with Robert Howlett to solve Springer's decomposition problem, resulting in the development of the ...