A group action is defined as a map from $G \times A \to A$, and it follows two properties. Now my basic question is:
How do we know that any arbitrary element $g$ acting on $a \in A$ wil give me an elemnet of $A$? What if $g.a $ is not in the set $A$, then is it not a group action?
I am sorry if the question is very trivial but could someone help me out with the definition of group action.
A group action is first of all a function. So, if $g \cdot a \notin A$ for some $g \in G$ and $a \in A$, then we don't have a function $G \times A \to A$ anymore, let alone a group action.
Answer to this question is actually we define an action in a way that it is a function first, then it satisfies the two properties you mentioned. So, if we have a case like $g \cdot a \notin A$ for some $g \in G$ and $a \in A$, we could extend $A$ to a bigger set $B$ so that $g\cdot a \in B$ and it satisfies the group action properties. But note that in this case, we must have an action $G \times B \to B$.