Let $A$ be a $G$ set, that is, a set acted upon by a group $G$ with action $\psi:G\times A\to A$ Then for any $g\in G$ the map $m_g\colon A\to A$ defined by $$m_g(x)= \psi(g,x)$$ is a permutation of $A$ (in other words, a bijective function from $A$ to itself) and so an element of $S_A$ We can even get an homomorphism from $G$ to $S_A$ by the rule $g\mapsto m_g$

If for any pair $g,h\in G$ $g\neq h$ we have $m_g\neq m_h$ in other words, the homomorphism $g\to m_g$ being injective, we say that the action is faithful.