# GA(2,2) is isomorphic to S4

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 general affine group of degree two over field:F2 (the field of two elements) is isomorphic to symmetric group:S4.

## Related facts

### Similar facts

- GA(1,3) is isomorphic to S3
- PGL(2,2) is isomorphic to S3
- PGL(2,3) is isomorphic to S4
- PGL(2,5) is isomorphic to S5

## Facts used

## Proof

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

1 | For any field the group has a faithful group action on and hence has an injective homomorphism to the symmetric group on . | By definition of , it has a faithful group action on . | ||

2 | For a field of size , has size . | Fact (1) | [SHOW MORE] | |

3 | For the field of size two, the symmetric group on is the symmetric group of degree four and its order is 24, and has order . | Step (2) | [SHOW MORE] | |

4 | For the field of size two, the injective homomorphism of Step (2) gives an isomorphism from to . | Steps (1), (3) | [SHOW MORE] |