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- Projective_linear_group abstract "In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient groupPGL(V) = GL(V)/Z(V)where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of the general linear group.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. Here SZ is the center of SL, and is naturally identified with the group of nth roots of unity in K (where n is the dimension of V and K is the base field).PGL and PSL are some of the fundamental groups of study, part of the so-called classical groups, and an element of PGL is called projective linear transformation, projective transformation or homography. If V is the n-dimensional vector space over a field F, namely V = Fn, the alternate notations PGL(n, F) and PSL(n, F) are also used.Note that PGL(n, F) and PSL(n, F) are equal if and only if every element of F has an nth root in F. As an example, note that PGL(2, C) = PSL(2, C), but PGL(2, R) > PSL(2, R); this corresponds to the real projective line being orientable, and the projective special linear group only being the orientation-preserving transformations.PGL and PSL can also be defined over a ring, with an important example being the modular group, PSL(2, Z).".
- Projective_linear_group thumbnail PSL-PGL.svg?width=300.
- Projective_linear_group wikiPageID "382667".
- Projective_linear_group wikiPageRevisionID "592858679".
- Projective_linear_group hasPhotoCollection Projective_linear_group.
- Projective_linear_group subject Category:Lie_groups.
- Projective_linear_group subject Category:Projective_geometry.
- Projective_linear_group type Abstraction100002137.
- Projective_linear_group type Group100031264.
- Projective_linear_group type LieGroups.
- Projective_linear_group comment "In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V).".
- Projective_linear_group label "Projectieve lineaire groep".
- Projective_linear_group label "Projective linear group".
- Projective_linear_group label "Проективная группа".
- Projective_linear_group label "射影線型群".
- Projective_linear_group label "射影线性群".
- Projective_linear_group sameAs Projektivní_grupa.
- Projective_linear_group sameAs 射影線型群.
- Projective_linear_group sameAs Projectieve_lineaire_groep.
- Projective_linear_group sameAs m.021t95.
- Projective_linear_group sameAs Q2997419.
- Projective_linear_group sameAs Q2997419.
- Projective_linear_group sameAs Projective_linear_group.
- Projective_linear_group wasDerivedFrom Projective_linear_group?oldid=592858679.
- Projective_linear_group depiction PSL-PGL.svg.
- Projective_linear_group isPrimaryTopicOf Projective_linear_group.
