Abstract:
We prove that if a finite simple linear or unitary group defined over a field of characteristic p and having dimension sufficiently large as compared with p acts on a finite-dimensional vector space over some field of the same characteristic p, then the corresponding semidirect product contains an element whose order is distinct from any element order of the simple group.
Citation:
A. V. Zavarnitsin, V. D. Mazurov, “On element orders in coverings of the simple groups Ln(q) and Un(q)”, Группы и графы, Trudy Inst. Mat. i Mekh. UrO RAN, 13, no. 1, 2007, 89–98; Proc. Steklov Inst. Math. (Suppl.), 257, suppl. 1 (2007), S145–S154
\Bibitem{ZavMaz07}
\by A.~V.~Zavarnitsin, V.~D.~Mazurov
\paper On element orders in coverings of the simple groups $L_n(q)$ and $U_n(q)$
\inbook Группы и графы
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2007
\vol 13
\issue 1
\pages 89--98
\mathnet{http://mi.mathnet.ru/timm73}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2338246}
\elib{https://elibrary.ru/item.asp?id=12040754}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2007
\vol 257
\issue , suppl. 1
\pages S145--S154
\crossref{https://doi.org/10.1134/S0081543807050100}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34547655629}
Linking options:
https://www.mathnet.ru/eng/timm73
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This publication is cited in the following 14 articles:
Nanying Yang, Ilya Gorshkov, Alexey Staroletov, Andrey V. Vasil'ev, “On recognition of direct powers of finite simple linear groups by spectrum”, Annali di Matematica, 202:6 (2023), 2699
V. I. Zenkov, “Intersections of two nilpotent subgroups in finite groups with socle $L_2(q)$”, Siberian Math. J., 57:6 (2016), 1002–1010
A. M. Staroletov, “On recognition by spectrum of the simple groups $B_3(q)$, $C_3(q)$, and $D_4(q)$”, Siberian Math. J., 53:3 (2012), 532–538
Khosravi B., Moradi H., “Quasirecognition by prime graph of finite simple groups $\mathrm{L}_n(2)$ and $\mathrm{U}_n(2)$”, Acta Math. Hungar., 132:1-2 (2011), 140–153
Darafsheh M.R., “Recognition of the projective special linear group over GF(3)”, Acta Math. Sin. (Engl. Ser.), 26:3 (2010), 477–488
I. B. Gorshkov, “Raspoznavanie po spektru konechnykh prostykh grupp, prostye deliteli poryadkov kotorykh ne prevoskhodyat $17$”, Sib. elektron. matem. izv., 7 (2010), 14–20
Khosravi B., “Some characterizations of $L^9(2)$ related to its prime graph”, Publ. Math. Debrecen, 75:3-4 (2009), 375–385
V. D. Mazurov, G. Y. Chen, “Recognizability of finite simple groups $L_4(2^m)$ and $U_4(2^m)$ by spectrum”, Algebra and Logic, 47:1 (2008), 49–55
M. A. Grechkoseeva, “Recognition by spectrum for finite linear groups over fields of characteristic 2”, Algebra and Logic, 47:4 (2008), 229–241
A. V. Zavarnitsine, “Properties of element orders in covers for $\operatorname L_n(q)$ and $\operatorname U_n(q)$”, Siberian Math. J., 49:2 (2008), 246–256
Grechkoseeva M.A., Shi Wujie, Vasilev A.V., “Recognition by spectrum for finite simple groups of Lie type”, Front. Math. China, 3:2 (2008), 275–285
Zavarnitsine A.V., “On recognition by spectrum among covers of finite simple unitary and linear groups”, Dokl. Math., 78:1 (2008), 481–484
A. V. Zavarnitsine, “Exceptional action of the simple groups $L_4(q)$ in the defining characteristic”, Sib. elektron. matem. izv., 5 (2008), 68–74
Suprunenko I.D., Zalesski A.E., “Fixed vectors for elements in modules for algebraic groups”, Internat. J. Algebra Comput., 17:5–6 (2007), 1249–1261
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