Let $G$ be a group, $F$ a field, and $V$ be an $F[G]$ module (equivalently $F$-representation of $G$). The following definition is well-known.
Definition 1. We say that $V$ is irreducible (or simple $F[G]$-module) if there is no proper non-zero subspace (submodule) $W$ of $V$ such that $g.W\subseteq W$ $\forall g\in G$.
Question: Can we reformulate this definition in the following way?
Definition 2. We say that $V$ is irreducible (or simple $F[G]$-module) if for some $0\neq v\in V$, $\langle g.v \,\colon g\in G\rangle =V$.