• Corpus ID: 227247873

On non-surjective word maps on $\mathrm{PSL}_{2}(\mathbb{F}_{q})$.

@article{Biswas2020OnNW,
  title={On non-surjective word maps on \$\mathrm\{PSL\}\_\{2\}(\mathbb\{F\}\_\{q\})\$.},
  author={Arindam Biswas and Jyoti Prakash Saha},
  journal={arXiv: Group Theory},
  year={2020}
}
Jambor--Liebeck--O'Brien showed that there exist non-proper-power word maps which are not surjective on $\mathrm{PSL}_{2}(\mathbb{F}_{q})$ for infinitely many $q$. This provided the first counterexamples to a conjecture of Shalev which stated that if a two-variable word is not a proper power of a non-trivial word, then the corresponding word map is surjective on $\mathrm{PSL}_2(\mathbb{F}_{q})$ for all sufficiently large $q$. Motivated by their work, we construct new examples of these types of… 

References

SHOWING 1-10 OF 16 REFERENCES

Word maps, conjugacy classes, and a noncommutative Waring-type theorem

Let w = w(x\,..., Xd) 1 be a nontrivial group word. We show that if G is a sufficiently large finite simple group, then every element g e G can be expressed as a product of three values of w in G.

Word maps and Waring type problems

Waring's problem asks whether every natural number is a sum of g(k) fcth powers (where g is a suitable function). This was solved affirmatively by Hubert in 1909. Optimizing g(k) has been a central

On the construction of Galois extensions of function fields and number fields

This paper consists of two parts and an appendix. In Part 1, we investigate Galois converings and consider the problem of reducing their fields of definition. We restrict ourselves to PSL 2

Diameters of finite simple groups: sharp bounds and applications

Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph r(G, S) associated with G and S, up to a multiplicative constant. Many applications

The Ore conjecture

The Ore conjecture, posed in 1951, states that every element of every finite non-abelian simple group is a commutator. Despite considerable effort, it remains open for various infinite families of

Some word maps that are non‐surjective on infinitely many finite simple groups

We provide the first examples of words in the free group of rank 2 that are not proper powers and for which the corresponding word maps are non‐surjective on an infinite family of finite non‐abelian

Characters of Free Groups Represented in the Two-Dimensional Special Linear Group*

We consider here the problem of determining when two elements in a free group will have the same character under all possible representations of the given group in the special linear group of 2 x 2

The Waring problem for finite simple groups

The classical Waring problem deals with expressing every natural number as a sum of g(k) k-th powers. Recently there has been considerable interest in similar questions for non-abelian groups, and