Prologue~
'To be, or not to be, that is the question...'
~William Shakespeare (Hamlet)
But the deeper question in mathematics is:
To act or not to act~ and if so, how faithfully?
In this post, we deal with group actions, which would describe how a group acts or moves the elements of a set. Some movements are clear and precise, while others slightly smudged.
Verse I~ The call for an action.
Consider a group $G$ and a set $X$, and let us try defining a map from the group to the set.
$$\phi : G \times X \rightarrow X$$
given by $(g,x) \mapsto g \cdot x$ (this is a left group action), satisfying $e \cdot x=x$ and $$(gh) \cdot x= g \cdot (h \cdot x)$$
This is equivalent to a homomorphism $\phi : G \rightarrow Sym(X)$, where the group reveals itself through permutations.
Here, an element from $G$ acts on an element from $X$ or maps this function composition to an element in $X$ again.
Verse II~ To be faithful?
One can begin to ponder- what if there are is exactly one element in $G$ which acts on an element $x \in X$ to give out some element in $y \in X$? This is faithfulness.
An action like this can also be referred to as unambiguous or faithful.
Equivalently, $$ker\ \phi =\{ g \in G \ | g \cdot x=x \ \forall \ x \in X\}= \{e\}.$$
The faithfulness of an action ensures that no non-identity element can fix all the elements of $X$, i.e. to say that since a group action is about moving things around, no non-identity element can kill the motion (or the vibe!). This implies that $G$ embeds into $Sym(X)$, and its structure is perfectly mirrored in the symmetries of $X$.
Unambiguous (or simply transitive) actions are both transitive and free, where transitive implies that an element $ x \in X$ consists of a single orbit under the action of $G$, and free implies that no non-identity element fixes any element in the set $X$. Transitivity is a global condition and demands kinship.
Verse III~ To be not!
But, what if the map $\phi$ fails to be injective? This seems to be a depiction of unfaithfulness!
An action like this would then be called ambiguous or unfaithful (or non-faithful), and the kernel of such an action is then non-trivial.
Ambiguous actions are either non-free, or non-transitive, or neither free nor transitive hence, there can exist multiple non identity elements which fix at least one point in $X$, this hampers the uniqueness and hence the action is not free and an unfaithful one.
Epilogue~
The kernel is the silent pause—unappreciated, yet defining the rhythm of the action, the quiet moments between the notes.
To be faithful is to be truly representative—to let every actor speak their line distinctly—no voice muted, no gesture wasted.
And unfaithfulness is the unappreciated pauses in a theatrical narrative. What matters is the appreciation of the pauses, the transitions and representation, and a faithful action preserves that.
To be unfaithful is to be suppressive. Unfaithful actions choke the performance - these are actors who forget their lines ever so often. It is a smudged lens and the eyes of the beholder find it to be difficult to see beyond and interpret the movements on stage and the eyes long for clarity.
Whereas, transitivity is the choreography that carries each dancer across the stage, while freeness is their unconstrained motion—but faithfulness is the lighting that lets us see every step clearly. Without it, we glimpse only fragments of the dance.