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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 infinite general linear group or stable general linear group is the direct limit of the inclusions GL(n, F) → GL(n + 1, F) as the upper left block matrix. It is denoted by either GL( F ) or GL(∞, F ) , and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places.
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
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)
The group GL n (K) itself; The special linear group SL n (K) (the subgroup of matrices with determinant 1); The group of invertible upper (or lower) triangular matrices; If g i is a collection of elements in GL n (K) indexed by a set I, then the subgroup generated by the g i is a linear group.
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.
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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 .
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