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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
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.
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
The generator of any continuous symmetry implied by Noether's theorem, the generators of a Lie group being a special case. In this case, a generator is sometimes called a charge or Noether charge, examples include: angular momentum as the generator of rotations, [3] linear momentum as the generator of translations, [3]
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 projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly: PSL(V) = SL(V) / SZ(V) where SL(V) is the special linear group over V and SZ(V) is the subgroup of scalar transformations with unit determinant.
In mathematics, the special linear Lie algebra of order over a field, denoted or (,), is the Lie algebra of all the matrices (with entries in ) with trace zero and with the Lie bracket [,]:= given by the commutator. This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras.
The second row is the same generator with a seed of 3, which produces a cycle of length 2. Using a = 4 and c = 1 (bottom row) gives a cycle length of 9 with any seed in [0, 8]. A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation.