Motion and distinguishing number two - Conder, Marston D. E. (Ars mathematica contemporanea, 2011)
A group ?$A$? acting faithfully on a finite set ?$X$? is said to have distinguishing number two if there is a proper subset ?$Y$? whose (setwise) stabilizer is trivial. The motion of ?$A$? acting on ?$X$? is defined as the largest integer ?$k$? such that all non-trivial elements of ?$A$? move at least ?$k$? elements of ?$X$?. The Motion Lemma of Russell and Sundaram states that if the motion is at least ?$2 \log_2 |A|$?, then the action has distinguishing number two. When ?$X$? is a vector space, group, or map, the Motion Lemma and elementary estimates of the motion together show that in all but finitely many cases, the action of Aut?$(X)$? on ?$X$? has distinguishing number two. A new lower bound for the motion of any transitive action gives similar results for transitive actions with restricted point-stabilizers. As an instance of what can happen with intransitive actions, it is shown that if ?$X$? is a set of points on a closed surface of genus ?$g$?, and ?$|X|$? is sufficiently large with respect to ?$g$?, then any action on ?$X$? by a finite group of surface homeomorphisms has distinguishing number two.
Keywords: delovanje grupe; distinguishing number; gibanje; group action; motion; razlikovalno število; stabilizator; stabilizer