I may be missing something important , but i don't understand why group action and G modules are using the same notation , even in some texts i feel like some authors call "action" a G module that is not precised to be a representation of permutation : What i am mainly refering to is :
Let's define $(V,p)$ a G-module. Then , p(g)(v) is noted g.v , and i find it completely illegible as for me group actions were already using this main notation , and that for me group actions and a (non permutation representation) G module are not the same object.
What am i missing ?
If $\rho:G\to GL(V)$ is a representation, then the map $G\times V\to V$ defined by $(g,v)\mapsto \rho(g)v$ is a group action, so it's perfectly suitable to adopt the conventional shorthand for group actions. Even if it isn't a permutation representation (that is, it doesn't come from an action of $G$ on a basis), it is still technically permuting the elements of the set $V$, so it is a group action! For this reasons, representations are often called linear actions.