Let $X$ be a finite set and let $G$ be a group. What is the meaning of $G$ being a "transitive permutation" on the set $X$?
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$\begingroup$You say $X$ is a finite set, not a group. Thus "on group $X$" (quoting your question as it now stands) is wrong. One may say $G$ is a transitive permutation group on the set $X$, but not on the group $X$ unless $X$ is a group.
It makes sense to speak of a transitive permutation group on a set $X$, but one does not say "$G$ is a transitive permutation". It is the permutation group that is transitive on the set $X$; the thing that is transitive is not a permutation. Your subject line as it now stands asks about a "transitive permutation". Again, it is not the permutation, but the group of permutations, that is transitive on the set $X$.
Parse the phrase like this:
$G$ is a transitive $\left\{\text{permutation group}\right\}$ on $X$,
not like this:
$G$ is a $\left\{\text{transitive permutation}\right\}$ group on $X$.
That $G$ is a transitive permutation group on $X$ means that for every pair $x,y\in X$, there is some permutation $g$ in the group $G$ that moves $x$ to $y$.
For example, the group of all shifts parallel to the $x$-axis in the $(x,y)$-plane is not transitive on the plane because there is no shift parallel to the $x$-axis that moves $(0,0)$ to $(1,1)$. On the other hand, the group of all translations of the plane is transitive on the plane.
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