What does it mean to be a transitive permutation group?

Question

Let $X$ be a finite set and let $G$ be a group. What is the meaning of $G$ being a "transitive permutation group" on the set $X$? (note: this question and title have been edited with the correct parsing)

Answer 1

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.

Answer 2

It means that the subgroup $G\le S_X:=\operatorname{Sym}(X)$ still acts transitively on $X$ as a group of permutations ("naturally").

Examples. For $|X|\ge 2$, the trivial one is the whole $S_X$, as for every $x,y\in X$ the transposition $(xy)$ does the job. For $|X|\ge 4$, the alternating subgroup $A_X$ is transitive, as for every $x,y\in X$ the double transposition $(xy)(zw)$, for some $z,w\in X$, does the job. By Cayley's theorem, and because there are cyclic groups of every order, $S_X$ always has transitive subgroups of order $|X|$.

Counter-example. Any pointwise stabilizer is not transitive, as by definition no one element of it can move the fixed point wheresoever.

Answer 3

Answering simply: A permutation group G is transitive over a finite set X if for every x,y in X, there is a g in G such that y=gx, i.e. there is a g that maps x to y.