Search results
Results from the WOW.Com Content Network
The general linear group GL(2, 7) consists of all invertible 2×2 matrices over F 7, the finite field with 7 elements. These have nonzero determinant. The subgroup SL(2, 7) consists of all such matrices with unit determinant. Then PSL(2, 7) is defined to be the quotient group. SL(2, 7) / {I, −I} obtained by identifying I and −I, where I is ...
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 (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.
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.
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.
Reductive groups include the most important linear algebraic groups in practice, such as the classical groups: GL(n), SL(n), the orthogonal groups SO(n) and the symplectic groups Sp(2n). On the other hand, the definition of reductive groups is quite "negative", and it is not clear that one can expect to say much about them.
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.
These groups are usually classified by some typical normal subgroup, this normal subgroup is denoted by G 0 and are written in the third column of the table. The notation 2 1+4 − stands for the extraspecial group of minus type of order 32 (i.e. the extraspecial group of order 32 with an odd number (namely one) of quaternion factor).