Geodesic
Exceptional action of the simple groups $L_4(q)$ in the defining characteristic
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 5 (2008), pp. 68-74.

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We find a counterexample to Problem 14.60 from the Kourovka notebook by giving an example of a group $\rm {PSL}_4(q)$ which is not recognizable among its covers from spectrum. Namely, we show that such a group has a module in the defining characteristic with the property that the element order set of the corresponding semidirect product equals that of the group itself. 
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A. V. Zavarnitsine. Exceptional action of the simple groups $L_4(q)$ in the defining characteristic. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 5 (2008), pp. 68-74. http://geodesic.mathdoc.fr/item/SEMR_2008_5_a7/

[1] “Unsolved problems in group theory”, The Kourovka notebook, 14th ed., Sobolev Inst. Mat., Novosibirsk, 1999 | MR

[2] A. V. Zavarnitsine, “Properties of element orders in covers for $\rm L_n(q)$ and $\rm U_n(q)$”, Sib. Math. J., 49:2 (2008), 246–256 | DOI | MR | Zbl

[3] A. A. Buturlakin, “The spectra of the linear and unitary groups”, Algebra and Logic, 47:2 (2008) (to appear) | DOI | MR | Zbl

[4] V. D. Mazurov, A. V. Zavarnitsine, “On element orders in coverings of the simple groups $\rm L_n(q)$ and $\rm U_n(q)$”, Proc. Inst. Math. Mech., 257, suppl. 1 (2007), S145–S154 | MR

[5] R. P. Brent, P. L. Montgomery, H. J. J. te Riele, Factorizations of Cunningham numbers with bases 13 to 99: Millennium Edition, Modelling, Analysis and Simulation, MAS-R0107, Amsterdam, 2001, viii+20 pp.

[6] R. Steinberg, Lectures on Chevalley groups, Yale University, New Haven, 1968 | MR | Zbl

[7] A. A. Buturlakin, M. A. Grechkoseeva, “The cyclic structure of maximal tori of the finite classical groups”, Algebra and Logic, 46:2 (2007), 73–89 | DOI | MR | Zbl

[8] A. V. Zavarnitsine, “Recognition of the simple groups $U_3(q)$ by element orders”, Algebra and Logic, 45:2 (2006), 106–116 | DOI | MR | Zbl