### Motion and distinguishing number two

#### Abstract

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

Distinguishing number, group action, stabilizer, motion

DOI: https://doi.org/10.26493/1855-3974.192.531

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications