GA(2,2) is isomorphic to S4

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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

  1. 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]
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]
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]