It shows that the axiomatization of transformation groups is correct. Any transformation group is an axiomatic group, and any axiomatic group -- a structure, or more precisely a "model", satisfying the group axioms -- is a group of transformations. Historically, concrete groups of transformation appeared before abstract groups defined by algebraic axioms, so there is a question whether the algebra captures all (or possibly, more than) the intended examples.

Also, Cayley's theorem with its initially strange but retroactively instinctive idea of considering an object's internal "action on itself", is an early example of the formal style typical of later algebra and it helps prepare the ground psychologically for working in an abstract or axiomatized mode that was new in the 19th century.

It shows that the axiomatization of transformation groups is correct. Any transformation group is an axiomatic group, and any axiomatic group -- a structure, or more precisely a "model", satisfying the group axioms -- is a group of transformations. Historically, concrete groups of transformation appeared before abstract groups defined by algebraic axioms, so there is a question whether the algebra captures all (or possibly, more than) the intended examples.

Also, Cayley's theorem with its initially strange but retroactively instinctive idea of considering a structure's internal "action on itself", promoting symmetries of an object to the status of an object in their own (collective) right, is an early example of the formal style typical of later algebra and it helps prepare the ground psychologically for working in an abstract or axiomatized mode that was new in the 19th century.