# PGL(2,3) is isomorphic to S4

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

## Related facts

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

The proof technique shows that there is an injective homomorphism from $PGL(2,q)$ (order $q^3 - q$) into $S_{q+1}$ (order $(q + 1)!$). In the cases $q = 2$ ($PGL(2,2) \to S_3$) and $q = 3$ ($PGL(2,3) \to S_4$) the orders on both sides are equal, hence the injective homomorphism is an isomorphism. For $q \ge 4$, the injective homomorphism is not an isomorphism.

## Proof

Step no. Assertion/construction Facts used Previous steps used Explanation
1 For any field $k$, $GL(2,k)$ has a natural action on the set $\mathbb{P}^1(k)$ of all lines through the origin in $k^2$. 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 $PGL(2,k)$ on the set $\operatorname{P}^1(k)$ of all lines through the origin in $k^2$, and hence an injective homomorphism from $PGL(2,k)$ to the symmetric group on the set $\mathbb{P}^1(k)$. Fact (1) Step (1) [SHOW MORE]
3 For $k$ a finite field of size $q$, $\mathbb{P}^1(k)$ has size $q + 1$. Fact (2) [SHOW MORE]
4 For $k$ a finite field of size $q$, $PGL(2,k)$, also denoted $PGL(2,q)$ has order $q^3 - q$. Fact (3) [SHOW MORE]
5 For $k$ the field of size three, we have $\operatorname{Sym}(\mathbb{P}^1(k)) = S_4$ and its order is $24$ and $|PGL(2,k)| = |PGL(2,3)| = 24$ Steps (3), (4) [SHOW MORE]
6 For $k$ the field of size three, the injective homomorphism of Step (2) gives an isomorphism from $PGL(2,3)$ to $S_4$. 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 $PGL(2,3)$ is "S4" meaning that it is the symmetric group of degree four. NA