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https://en.wikipedia.org/wiki/Projective_linear_group
Projective linear group
(抜粋)
PGL(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

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.

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.

there are other exceptional isomorphisms between projective special linear groups and alternating groups (these groups are all simple, as the alternating group over 5 or more letters is simple):

L_2(4) ＝〜 A_5
L_2(5) ＝〜 A_5 (see here for a proof)

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