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Projective linear group
- PGL(V) = GL(V)/Z(V)
where GL(V) is the general linear group on V and Z(V) is the group of all nonzero scalar transformations of V.
The projective special linear group is defined analogously:
- PSL(V) = SL(V)/SZ(V)
where SZ(V) is the group of scalar transformations with unit determinant.
Note that the groups Z(V) and SZ(V) are the centers of GL(V) and SL(V) respectively. If V is an n-dimensional vector space over a field F the alternate notations PGL(n, F) and PSL(n, F) are also used.
The name comes from projective geometry, where the projective group acting on homogeneous coordinates (x0:x1: … :xn) is the underlying group of the geometry (N.B. this is therefore PGL(n+1, F) for projective space of dimension n). Stated differently, the natural action of GL(V) on V descends to an action of PGL(V) on the projective space P(V).
The projective special linear groups PSL(n,Fq) for a finite field Fq are often written as PSL(n,q) or Ln(q). They are finite simple groups whenever n is at least 2, except for L2(2) (which is isomorphic to the symmetric group on 3 letters) and L2(3) (which is isomorphic to the alternating group on 4 letters).
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