PSL(2,7) is isomorphic to PSL(3,2)

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This article gives a proof/explanation of the equivalence of multiple definitions for the term projective special linear group:PSL(3,2)
View a complete list of pages giving proofs of equivalence of definitions

Statement

First, note that, by the isomorphism between linear groups over field:F2, we have:

PSL(3,2) \cong PGL(3,2) \cong SL(3,2) \cong GL(3,2)

Here, PSL(3,2) stands for the projective special linear group of degree three over field:F2. PGL stands for the projective general linear group of degree three, SL stands for the special linear group of degree three, and GL stands for the general linear group of degree three.

The claim is that all of these isomorphic groups are isomorphic to the projective special linear group of degree two over field:F7, denoted PSL(2,7).