permutation module

Question

If X is a G-set, then the free abelian group Z[X] generated by X and extends the action of G on X to a Z-linear action of G on Z[X]. The resulting G-module is called a Permutation module. (Kenneth S. Brown, Cohomology of Groups, pg 13) What does it meant by Z-linear action and how does permutation module looks like? Please explain

Answer 1

Linear extension means $g\cdot\sum_{x\in X} z_xx = \sum_{x\in X}z_x(g\cdot x)$, where $z_x\in\mathbb Z$. Careful if $X$ is infinite.