Characteristic covering numbers of finite simple groups

Abstract

In the past few decades there has been considerable interest in word maps on groups with emphasis on (non-abelian) finite simple groups. Various asymptotic results (holding for sufficiently large groups) have been obtained. More recently non-asymptotic results (holding for all finite simple groups) emerged, with emphasis on particular words (commutators and certain power words) which are not an identity of any finite simple group. In this paper we initiate a systematic study of all words with the above property. In particular, we show that, if \(w_1, \ldots , w_6\) are words which are not an identity of any (non-abelian) finite simple group, then \(w_1(G)w_2(G) \ldots w_6(G) = G\) for all (non-abelian) finite simple groups G. Consequently, for every word w, either \(w(G)^6 = G\) for all finite simple groups, or \(w(G)=1\) for some finite simple group. These theorems follow from more general results we obtain on characteristic collections of finite groups and their covering numbers, which are of independent interest and have additional applications.