PSL(2,5) is isomorphic to A5

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This article gives a proof/explanation of the equivalence of multiple definitions for the term alternating group:A5
View a complete list of pages giving proofs of equivalence of definitions

Statement

The projective special linear group of degree two over field:F5, which we denote PSL(2,5), is isomorphic to alternating group:A5.

Facts used

  1. A_5 has order 5!/2 = 60
  2. PSL(2,5) has order (5^3 - 5)/2 = 60 by order formulas for linear groups of degree two
  3. A5 is the unique simple non-abelian group of smallest order: This says that A_5 is the simple non-abelian group of smallest possible order and also that any simple non-abelian group of the same order as A_5 must be isomorphic to A_5.
  4. Projective special linear groups are simple with some exceptions, but PSL(2,5) is not among the exceptions, so PSL(2,5) is simple.

Proof

The proof follows directly by combining Facts (1)-(4).