If a group $G$ is acting on another group $N$, $g \in G$ and $m,n \in N$,is it true that $g\cdot(mn)= (g\cdot m)(g\cdot n)$ where $\cdot$ represents the group action and $mn$ represents the product in the group $N$.
I am trying to verify that the group product in a semidirect product is associative and seems to rely on this fact, but I can't see why that is true from the definition after playing around with it for a little while.
Very often, groups act on objects with some structure by structure preserving maps. For example, a group may act on a topological space by homeomorphisms; or on a ring by ring automorphisms; or on another group by group automorphisms. If a group acts on another group by group automorphisms, that means precisely that $g\cdot (m n) = (g\cdot m) (g\cdot n)$. Put another way, an action $\alpha$ of $G$ on $N$ by automorphisms is a homomorphism $\alpha: G \to \text{Aut}(N)$.
In the definition of a semi-direct product of groups, the group which acts is required to act by automorphisms on the acted-upon group. This cannot be derived from something else, it is a requirement. Go back to your text or lecture notes, or the discussion of semi-direct products in some reference work, and you will see that is the case.