PGL(2,5) is isomorphic to S5

This article gives a proof/explanation of the equivalence of multiple definitions for the term symmetric group:S5
View a complete list of pages giving proofs of equivalence of definitions

Statement

The projective general linear group of degree two over field:F5, denoted $PGL(2,5)$, is isomorphic to symmetric group:S5.

Related facts

Similar facts

• PSL(2,5) is isomorphic to A5
• PGammaL(2,4) is isomorphic to S5
• PSL(2,4) is isomorphic to A5
• PGL(2,3) is isomorphic to S4
• PSL(2,3) is isomorphic to A4
• PSL(2,9) is isomorphic to A6

Facts used

1. PSL(2,5) is isomorphic to A5
2. $S_5$ is the automorphism group of $A_5$
3. $PGL(2,p)$ is the automorphism group of $PSL(2,p)$ for any prime $p$

Proof

The proof follows directly by combining Facts (1)-(3).