Cayley's Theorem via Regular Action

Abstract Algebra · Axiom Academy

EXAMPLE Cayley's Theorem via Regular Action Proving every group embeds in a symmetric group through the example of Z₄ Excellent work! You've constructed a proof of Cayley's Theorem using the regular action. Here's what we learned: Cayley's Theorem is constructive: We don't just know the embedding exists—we can build it explicitly using the regular action Regular action λ(g)(x) = gx: Each group element g acts by left multiplication, creating a permutation of the group Homomorphism property: λ(gh) = λ(g)∘λ(h) ensures the group structure is preserved Injectivity is key: Different group elements produce different permutations, making λ one-to-one Universal result: This works for ANY group—finite or infinite, abelian or non-abelian Cayley's Theorem shows that abstract groups are not so abstract after all—every group can be realized concretely as symmetries (permutations). This fundamental result connects abstract algebra to the concrete world of transformations!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.