Relation between group action in abstract algebra and semi-automaton If $G$ is a set acting on a set $A$, can it define a semi-automaton? How is this concept of abstract algebra related to semi-automata?
 A: Your notation is a bit misleading, since $A$ usually denotes an alphabet. So let me switch the role of $G$ and $A$. Suppose that $A$ acts on $G$, that is, there is a map  $(g,a) \mapsto g \cdot a$ from $G \times A$ to $G$. This map can be extended, to an action on $G$ of every word $u$ of $A^*$. This can be done by induction on the length of $u$. Let $g \in G$. For the empty word $1$, just set $g \cdot 1 = g$. Now if $u$ is a word and $a$ is a letter, set $g \cdot (ua) = (g \cdot u) \cdot a.\ $ You define in this way a semi-automaton $(G, A)$ where $G$ is the set of states and $A$ is the alphabet.
A: Guess by $G$ you mean a group, and not just as a set, as written in the title?
Then for a group $G$ and a set $A$ a group action is a mapping $A \times G \to G$ such that $a1 = a$ and $a(gh) = (ag)h$ for $a \in A$ and $g,h \in G$.
On the other side, a semi-automaton is a certain computational device to decribe formal languages or (sequentiell) computations. Without giving all the definitions, the more algebraic definition of a (finite) semi-automaton are two finite sets $Q$ (states) and $\Sigma$ (alphabets) such that $\Sigma$ acts on $Q$, i.e. we have a mapping $Q \times \Sigma \to Q$.
Now looking at the free monoid $\Sigma^{\ast}$ this gives us a action of the free monoid $\Sigma^{\ast}$ on $Q$. This is what Jean-Eric Pin wrote in his answer (he took $A$ for the alphabet and $G$ for the state set, I guess you mean $G$ acting on $A$ as written above, but the wording of your question is not clear about that...)
Anyway, are you asking about the relation of these two notions? They are closely related, but they are not the same, with different things that they should model in mind.
For a group action, it could be viewed as a (group-) automaton, the inputs are group elements and the state is the element of the permutation group after applying them to some initial permutation. Think of symmetry groups, for example the symmetry of a pentagon. The current configuration is a state, and applying some group element changes it. And if you want to push this further every group element could be written as a word over some alphabet (the generators and their inverses) and by this you get even an action of words on your set.
But in general a group action is looked at as given and fixed, and we try to classify groups that could act in a certain way, try to determine point stabilizers etc. On the other side, for an automaton we look at word sets, call certain states initial or final, and try to minimize them, i.e. try to find automata accomplishing the same (external) behaviour with fewer states. To my knowledge such questions are never asked for a group action, because the focus is different. For example in the automata setting it is quite often asked what words transform certain states into each other and the focus is on the resulting set of words. In the group action setting the group elements transforming two points into each other would be somehow analog to that, and this would be a coset of the point stabilizer, which has entirely a different structure as a formal language and I would think totally different about them (despite the mentioned connections above).
