After learning how homomorphisms compare two groups through structure-preserving maps, we now ask a different but equally powerful question: how can one group move the elements of a set? This is the starting point of group actions and G-Sets. A group action translates an abstract group into visible transformations of a set, so that algebraic multiplication becomes movement, symmetry, or rearrangement. Students often think that a group action is only a formula, but the essential point is compatibility: multiplying first in the group must give the same result as acting step by step. In this lesson we shall define left actions, introduce $G$-sets, and verify the standard actions coming from permutations, conjugation, and matrices.

Let $G$ be a group with identity element $e$, and let $S$ be a non-empty set. A \textbf{left action} of $G$ on $S$ is a function $$\begin{equation*} G\times S\to S \end{equation*}$$ written as $$\begin{equation*} (g,x)\mapsto g\cdot x \end{equation*}$$ such that the following conditions hold for all $g_1,g_2\in G$ and all $x\in S$: (i) $(g_1g_2)\cdot x=g_1\cdot(g_2\cdot x)$, (ii) $e\cdot x=x$.