On the Surjectivity of Engel Words on PSL(2,q)

TL;DR

This paper studies the surjectivity of the n-th Engel word map on PSL(2,q) and SL(2,q), establishing conditions under which it is surjective and almost measure preserving, with implications for finite simple groups.

Contribution

It proves the surjectivity of the Engel word map on PSL(2,q) for large q and confirms the conjecture for small n, advancing understanding of word maps in finite groups.

Findings

Engel word map is surjective on PSL(2,q) for q > Q(n).

For n<5, the map is surjective for all PSL(2,q).

The map is almost measure preserving for PSL(2,q) with odd q.

Abstract

We investigate the surjectivity of the word map defined by the n-th Engel word on the groups PSL(2,q) and SL(2,q). For SL(2,q), we show that this map is surjective onto the subset SL(2,q)\{-id} provided that q>Q(n) is sufficiently large. Moreover, we give an estimate for Q(n). We also present examples demonstrating that this does not hold for all q. We conclude that the n-th Engel word map is surjective for the groups PSL(2,q) when q>Q(n). By using the computer, we sharpen this result and show that for any n<5, the corresponding map is surjective for all the groups PSL(2,q). This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the n-th Engel word map is almost measure preserving for the family of groups PSL(2,q), with q odd, answering another question of Shalev. Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group SL(2,q).