Orblts under Actlons of Affine Groups over zyxwvutsrqponmlkjihgfedcbaZYXWV GF (2) Frederick Hoffman Department of Mathematics Florida Atlantic University Boca Raton, Florida 33432 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM Submitted by Hans Schneider ABSTRACT Let G be a group (or vector space) and A a group of transformations of G. A then acts as a group of transformations of P(G), the set of subsets of G. It is meaningful to study the orbit structure of P(G) under tbe action of A. The question of the existence of elements of P(G) with trivial isotropy subgroup seems to be of interest in studying tbe action of A on G. In this paper actions of affine groups over GF (2) are considered. It is proved, by an inductive construction, that every vector space over GF (2) of dimension at least six contains a subset with trivial isotropy subgroup. Let G be a group (or vector space) and A a group of transformations of G. A then acts as a group of transformations of P(G), the set of subsets of G. It is meaningful to study the orbit structure of P(G) under the action of A. The question of the existence of elements of P(G) with trivial isotropy subgroup seems to us to be of interest in studying the action of A on G. In a previous paper [l], Lloyd R. Welch and the author answered the question of what finite groups and vector spaces contain totally variant subsets (i.e., subsets moved by every non-identity automorphism, or in the vector space case, by every non-identity non-singular linear transformation). Here, actions of affine groups over GF (2) are considered. We prove that every vector space over GF (2) of dimension at least six contains an affinely totally variant subset. The cases for other finite fields are discussed in [3]. zyxwvutsrq DEFINITION . Let s be a subset of the vector space V over GF (2). If 5 has the property that the only non-singular linear transformation T on V with S T= S is the identity transformation, then S is totally variant (in V). If s has the property that the only affine transformation A on V with S A = 5 is the identity transformation, then S is affinely totally variant (in V). [An affine transformation A is one given by VA = uLA + wA for all v E V, where LINEAR ALGEBRA AND ITS APPLZCATZONS 13, 173-176 (1976) 0 American Elsevier Publishing Company,Inc., 1976 173