# PGammaL(2,4) is isomorphic to S5

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term symmetric group:S4

View a complete list of pages giving proofs of equivalence of definitions

## Statement

The projective semilinear group of degree two over field:F4 (the field of four elements) is isomorphic to symmetric group:S5.

In symbols:

.

## Related facts

### Similar facts

- PGL(2,5) is isomorphic to S5
- PSL(2,5) is isomorphic to A5
- PGL(2,4) is isomorphic to A5
- PGL(2,3) is isomorphic to S4
- PSL(2,3) is isomorphic to A4

## Facts used

- Equivalence of definitions of size of projective space
- Order formulas for linear groups of degree two

## Proof

Step no. | Assertion/construction | Facts used | Previous steps used | Explanation |
---|---|---|---|---|

1 | For any field , there is a natural faithful group action of on , the set of lines through the origin in , and hence an injective group homomorphism from to the symmetric group on . | Follows from definitions. | ||

2 | For the field of size , has size and thus the symmetric group on it has size . | Fact (1) | ||

3 | For the field of size with prime, has size . | Fact (2) | ||

4 | For the field of size , has order and is the symmetric group of degree and has order . | Steps (2), (3) | Plug in and evaluate. | |

5 | For the field of size , the homomorphism of Step (1) gives an isomorphism from to . | Steps (1), (4) | By Steps (1) and (4), we get an injective homomorphism from to . Again by Step (4), both groups are finite and have the same order, hence the injective homomorphism must be an isomorphism. |