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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 ...
PSL(2, 2) is isomorphic to the symmetric group S 3, and PSL(2, 3) is isomorphic to alternating group A 4. In fact, PSL(2, 7) is the second smallest nonabelian simple group, after the alternating group A 5 = PSL(2, 5) = PSL(2, 4). The number of conjugacy classes and irreducible representations is 6. The sizes of conjugacy classes are 1, 21, 42 ...
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.
L 2 (3) ≅ A 4 A 3 ≅ C 3 via the quotient by the Klein 4-group; L 2 (5) ≅ A 5. To construct such an isomorphism, one needs to consider the group L 2 (5) as a Galois group of a Galois cover a 5: X(5) → X(1) = P 1, where X(N) is a modular curve of level N. This cover is ramified at 12 points.
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.
SL(2, R) is the group of all linear transformations of R 2 that preserve oriented area. It is isomorphic to the symplectic group Sp(2, R) and the special unitary group SU(1, 1). It is also isomorphic to the group of unit-length coquaternions. The group SL ± (2, R) preserves unoriented area: it may reverse orientation.
In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. Linear algebra is the branch of mathematics concerning linear equations such as: