GA(2,2) 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 general affine group of degree two over field:F2 (the field of two elements) is isomorphic to symmetric group:S4.

Related facts

Similar facts

Facts used

Order formulas for linear groups of degree two

Proof

Step no.	Assertion/construction	Facts used	Previous steps used	Explanation
1	For any field $k$ the group $GA(2,k)$ has a faithful group action on $k^{2}$ and hence has an injective homomorphism to the symmetric group on $k^{2}$ .			By definition of $GA(n,k)$ , it has a faithful group action on $k^{n}$ .
2	For a field of size $q$ , $GA(2,q)=GA(2,\mathbb {F} _{q})$ has size $q^{2}(q^{2}-1)(q^{2}-q)$ .	Fact (1)		[SHOW MORE] Follows directly from Fact (1). Explicitly: $GA(2,q)$ is a semidirect product of $\mathbb {F} _{q}^{2}$ and $GL(2,q)$ . The former has order $q^{2}$ and the latter has order $(q^{2}-1)(q^{2}-q)$ , so $GA(2,q)$ has order $q^{2}(q^{2}-1)(q^{2}-q)$ .
3	For $k$ the field of size two, the symmetric group on $k^{2}$ is the symmetric group of degree four and its order is 24, and $GA(2,k)$ has order $2^{2}(2^{2}-1)(2^{2}-2)=24$ .		Step (2)	[SHOW MORE] The order of $GA(2,k)$ can be computed by the formula in Step (2). The size of $k^{2}$ is $2^{2}=4$ , so the symmetric group on it has order $4!=24$ .
4	For $k$ the field of size two, the injective homomorphism of Step (2) gives an isomorphism from $GA(2,2)$ to $S_{4}$ .		Steps (1), (3)	[SHOW MORE] By Step (1), there is an injective homomorphism from $GA(2,2)$ to $S_{4}$ . By Step (3), both groups have the same finite order, so the injective homomorphism is an isomorphism.