PGL(2,3) is isomorphic to S4

Statement
The projective general linear group of  degree two over field:F3 (the field of three elements) is isomorphic to symmetric group:S4.

Similar facts

 * PGL(2,2) is isomorphic to S3
 * GA(2,2) is isomorphic to S4
 * PGL(2,5) is isomorphic to S5

What the proof technique says for other projective general linear groups
The proof technique shows that there is an injective homomorphism from $$PGL(2,q)$$ (order $$q^3 - q$$) into $$S_{q+1}$$ (order $$(q + 1)!$$). In the cases $$q = 2$$ ($$PGL(2,2) \to S_3$$) and $$q = 3$$ ($$PGL(2,3) \to S_4$$) the orders on both sides are equal, hence the injective homomorphism is an isomorphism. For $$q \ge 4$$, the injective homomorphism is not an isomorphism.

Facts used

 * 1) uses::First isomorphism theorem
 * 2) uses::Equivalence of definitions of size of projective space
 * 3) uses::Order formulas for linear groups of degree two

GAP implementation
The fact that the groups are isomorphic can be tested in any of these ways: