Group actions can be transitive. Relations can be transitive.
Why do these concepts have the same name? Is there some relation that is naturally associated with a group action such that the relation is transitive if and only if the group action is transitive?
If not, who decided to use the word “transitive” for this property of group actions, and why? (I know the term appears in the Burnside finite groups book of 1911: “A permutation-group is called transitive when, by means of its permutations, a given symbol $a_1$ can be changed into every other symbol $a_2, a_3,\ldots, a_n$ operated on by the group.”)
For transitive graphs, the notion of “transtitivity” seems to be straightforwardly related to transitive group actions: A graph is vertex-transitive if its symmetry group acts transitively on the set of vertices. (And similarly edge-transitivity.) Did transitive graphs come first, and transitive group actions were a later generalization of that idea, or did transitive group actions come first, and then transitive graphs were an obvious special case?