Difference between revisions of "PGL(2,3) is isomorphic to S4"
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The [[fact about::projective general linear group;3| ]][[projective general linear group]] of [[fact about::projective general linear group of degree two;2| ]][[projective general linear group of degree two|degree two]] over [[field:F3]] (the field of three elements) is isomorphic to [[symmetric group:S4]]. | The [[fact about::projective general linear group;3| ]][[projective general linear group]] of [[fact about::projective general linear group of degree two;2| ]][[projective general linear group of degree two|degree two]] over [[field:F3]] (the field of three elements) is isomorphic to [[symmetric group:S4]]. | ||
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+ | ==Related facts== | ||
+ | ===Similar facts=== | ||
+ | |||
+ | * [[PGL(2,2) is isomorphic to S3]] | ||
+ | * [[GA(2,2) is isomorphic to S4]] | ||
+ | * [[PGL(2,5) is isomorphic to S5]] | ||
+ | |||
+ | ===What the proof technique says for other projective general linear groups=== | ||
+ | |||
+ | The proof technique shows that there is an injective homomorphism from <math>PGL(2,q)</math> (order <math>q^3 - q</math>) into <math>S_{q+1}</math> (order <math>(q + 1)!</math>). In the cases <math>q = 2</math> (<math>PGL(2,2) \to S_3</math>) and <math>q = 3</math> (<math>PGL(2,3) \to S_4</math>) the orders on both sides are equal, hence the injective homomorphism is an isomorphism. For <math>q \ge 4</math>, the injective homomorphism is ''not'' an isomorphism. | ||
==Facts used== | ==Facts used== |
Revision as of 18:03, 16 November 2011
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
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 . | [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 |
---|---|---|---|
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. |
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. |
StructureDescription(PGL(2,3)) | StructureDescription, PGL | "S4" | The structure description of is "S4" meaning that it is the symmetric group of degree four. |