scholarly journals Non-Hurwitz Classical Groups

2007 ◽  
Vol 10 ◽  
pp. 21-82 ◽  
Author(s):  
R. Vincent ◽  
A.E. Zalesski
Keyword(s):  
Finite Fields ◽  
Unitary Groups ◽  
Classical Groups ◽  
Special Linear ◽  
Low Dimensional

AbstractIn previous work by Di Martino, Tamburini and Zalesski [Comm. Algebra28 (2000) 5383–5404] it is shown that certain low-dimensional classical groups over finite fields are not Hurwitz. In this paper the list is extended by adding the special linear and special unitary groups in dimensions 8.9,11.13. We also show that all groups Sp(n, q) are not Hurwitz forqeven andn= 6,8,12,16. In the range 11 <n< 32 many of these groups are shown to be non-Hurwitz. In addition, we observe that PSp(6, 3),PΩ±(8, 3k),PΩ±10k), Ω(11,3k), Ω±(14,3k), Ω±(16,7k), Ω(n, 7k) forn= 9,11,13, PSp(8, 7k) are not Hurwitz.

scholarly journals CHARACTER LEVELS AND CHARACTER BOUNDS

2020 ◽  
Vol 8 ◽  
Author(s):  
ROBERT M. GURALNICK ◽  
MICHAEL LARSEN ◽  
PHAM HUU TIEP
Keyword(s):  
Random Walks ◽  
Mixing Time ◽  
Upper Bounds ◽  
Unitary Groups ◽  
Covering Number ◽  
Classical Groups ◽  
Dual Pairs ◽  
Special Linear ◽  
Exponential Character ◽  
Finite Classical Groups

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods.

scholarly journals A recognition algorithm for non-generic classical groups over finite fields

1999 ◽  
Vol 67 (2) ◽  
pp. 223-253 ◽  
Author(s):  
Azice C. Niemeyer ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Special Linear Group ◽  
Theoretical Background ◽  
Recognition Algorithm ◽  
Large Field ◽  
Matrix Group ◽  
Classical Groups ◽  
Original Algorithm ◽  
The Matrix

AbstractIn a previous paper the authors described an algorithm to determine whether a group of matrices over a finite field, generated by a given set of matrices, contains one of the classical groups or the special linear group. The algorithm was designed to work for all sufficiently large field sizes and dimensions of the matrix group. However, it did not apply to certain small cases. Here we present an algorithm to handle the remaining cases. The theoretical background of the algorithm presented in this paper is a substantial extension of that needed for the original algorithm.

scholarly journals THE -GENERATION OF THE SPECIAL LINEAR GROUPS OVER FINITE FIELDS

2016 ◽  
Vol 95 (1) ◽  
pp. 48-53 ◽  
Author(s):  
MARCO ANTONIO PELLEGRINI
Keyword(s):  
Finite Fields ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Special Linear ◽  
Special Linear Groups

We complete the classification of the finite special linear groups $\text{SL}_{n}(q)$ which are $(2,3)$-generated, that is, which are generated by an involution and an element of order $3$. This also gives the classification of the finite simple groups $\text{PSL}_{n}(q)$ which are $(2,3)$-generated.

On Shalev's conjecture for type 𝐴𝑛 and 2𝐴𝑛

2019 ◽  
Vol 22 (4) ◽  
pp. 713-728 ◽  
Author(s):  
Alexey Galt ◽  
Amit Kulshrestha ◽  
Anupam Singh ◽  
Evgeny Vdovin
Keyword(s):  
Unitary Groups ◽  
Special Linear

AbstractIn the paper, we consider images of finite simple projective special linear and unitary groups under power words. In particular, we show that, if {G\simeq\operatorname{PSL}_{n}^{\varepsilon}(q)}, then, for every power word of type {x^{M}}, there exist constants c and N such that {\lvert\omega(G)\rvert>c\frac{\ln(n)\lvert G\rvert}{n}} whenever {\lvert G\rvert>N}.

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