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 ![]() ![]() ![]() ![]() |
[SHOW MORE] | ||
2 | The action in Step (1) descends to a faithful group action of ![]() ![]() ![]() ![]() ![]() |
Fact (1) | Step (1) | [SHOW MORE] |
3 | For ![]() ![]() ![]() ![]() |
Fact (2) | [SHOW MORE] | |
4 | For ![]() ![]() ![]() ![]() ![]() |
Fact (3) | [SHOW MORE] | |
5 | For ![]() ![]() ![]() ![]() |
Steps (3), (4) | [SHOW MORE] | |
6 | For ![]() ![]() ![]() |
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 ![]() |
NA |