# PSL(2,5) is isomorphic to A5

From Groupprops

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 , is isomorphic to alternating group:A5.

## Facts used

- has order
- has order by order formulas for linear groups of degree two
- A5 is the unique simple non-abelian group of smallest order: This says that is the simple non-abelian group of smallest possible order and also that any simple non-abelian group of the same order as must be isomorphic to .
- Projective special linear groups are simple with some exceptions, but is not among the exceptions, so is simple.

## Proof

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