Group Actions Definition

Abstract Algebra · Axiom Academy

How groups can transform sets while preserving structure: a fundamental bridge between abstract algebra and geometry. Identity: for all , where e is the identity element of G We write (or simply gx ) to denote the result of the group element g acting on x . The compatibility axiom ensures that composing actions mirrors the group multiplication, while the identity axiom says the identity element acts trivially. 2. Example: Symmetric Group S_n The symmetric group S_n consists of all permutations of . It has a natural action on this set: a permutation simply rearranges the elements. This action satisfies both axioms: composing permutations corresponds to applying them successively, and the identity permutation leaves everything fixed. 3. Example: Conjugation Action Any group G can act on itself by conjugation . This is one of the most important actions in group theory, revealing internal symmetries of the group. 4. Example: Matrix Groups on Vector Spaces Matrix groups like (invertible matrices) naturally act on vector spaces by matrix multiplication. Matrix actions are fundamental in linear algebra and geometry: rotations, reflections, and scalings are all special cases of this general action. Let's verify that the conjugation action satisfies both axioms. This verification works similarly for all group actions.

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