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

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

2010-08-08

arxiv.org/abs/1008.1397v2

Mathematics

Group Theory

v2: 25 pages, minor changes, accepted to the Journal of Groups, Geometry and Dynamics

Scientific paper

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).

Affiliated with

Also associated with

No associations

Rating

On the Surjectivity of Engel Words on PSL(2,q) does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the Surjectivity of Engel Words on PSL(2,q), we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Surjectivity of Engel Words on PSL(2,q) will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-313322