# PGL(2,3) 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

## Contents

## Statement

The projective general linear group of degree two over field:F3 (the field of three elements) is isomorphic to symmetric group:S4.

## Related facts

### Similar facts

### What the proof technique says for other projective general linear groups

The proof technique shows that there is an injective homomorphism from (order ) into (order ). In the cases () and () the orders on both sides are equal, hence the injective homomorphism is an isomorphism. For , the injective homomorphism is *not* an isomorphism.

## Facts used

- First isomorphism theorem
- 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 , has a natural action on the set of all lines through the origin in . The kernel of this action is precisely the scalar matrices. | [SHOW MORE] | ||

2 | The action in Step (1) descends to a faithful group action of on the set of all lines through the origin in , and hence an injective homomorphism from to the symmetric group on the set . | Fact (1) | Step (1) | [SHOW MORE] |

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

4 | For a finite field of size , , also denoted has order . | Fact (3) | [SHOW MORE] | |

5 | For the field of size three, we have and its order is and | Steps (3), (4) | [SHOW MORE] | |

6 | For the field of size three, the injective homomorphism of Step (2) gives an isomorphism from to . | Steps (2), (5) | [SHOW MORE] |

## GAP implementation

The fact that the groups are isomorphic can be tested in any of these ways:

Command | Functions used | Output | Meaning | Memory usage |
---|---|---|---|---|

IsomorphismGroups(PGL(2,3),SymmetricGroup(4)) |
IsomorphismGroups, PGL, SymmetricGroup | [ (3,4), (1,2,4) ] -> [ (1,4), (1,2,3) ] |
The output is an actual mapping of generating sets for the groups (as they happen to be stored in GAP) that induces an isomorphism. The fact that a mapping is output indicates that an isomorphism does exist. If the groups were not isomorphic, an output of fail would be returned. |
22826 |

IdGroup(PGL(2,3)) = IdGroup(SymmetricGroup(4)) |
IdGroup, PGL, SymmetricGroup | true |
The ID of a group is uniquely determined by the group's isomorphism class, and non-isomorphic groups always have different IDs. This test can thus be used to check whether the groups are isomorphic. | NA |

StructureDescription(PGL(2,3)) |
StructureDescription, PGL | "S4" |
The structure description of is "S4" meaning that it is the symmetric group of degree four. | NA |