Projective linear group

From Academic Kids

The projective linear group of a vector space V over a field F is the quotient 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 linear groups therefore generalise the case PGL(2) of Möbius transformations (sometimes called the Möbius group), which acts on the projective line.

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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