MathGloss

A group action of a group $G$ on a set $X$ is a map $G\times A\to A$ such that

1. $g_1\cdot(g_2\cdot x)= (g_1g_2)\cdot x$ for all $g_1,g_2\in G$ and for all $x\in X$;
2. $e\cdot x = x$ for all $x\in X$.

Equivalently, we can describe a group action in terms of the homomorphism $\rho:G\to S_X$ (to the symmetric group on $X$) given by $\rho(g) = \sigma_g$ where $\sigma_g \in S_X$ is such that $x\mapsto g\cdot x$ under $\sigma_g$.

Wikidata ID: Q288465

This site is open source. Improve this page.