Abstract
The spectrum of a finite group is the set of its element orders. We give an affirmative answer to Problem 20.58(a) from the Kourovka Notebook proving that for every positive integer k, the k-th direct power of the simple linear group \(L_{n}(2)\) is uniquely determined by its spectrum in the class of finite groups provided n is a power of 2 greater than or equal to \(56k^2\).
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Bang, A.S.: Taltheoretiske Undersøgelser. Tidsskr. Math. 4(70–80), 130–137 (1886)
Bellotti, C., Keller, T.M., Trudgian, T.S.: New bounds for numbers of primes in element orders of finite groups, manuscript. arXiv:2211.05837
Buturlakin, A.A.: Spectra of finite linear and unitary groups. Algebra Log. 47(2), 91–99 (2008). https://doi.org/10.1007/s10469-008-9003-3
Buturlakin, A.A.: Spectra of finite symplectic and orthogonal groups. Sib. Adv. Math. 21(3), 176–210 (2011). https://doi.org/10.3103/S1055134411030035
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)
Gorshkov, I.B.: Characterization of groups with non-simple socle. Mediterr. J. Math 19(2), Paper No. 56 (2022). https://doi.org/10.1007/s00009-022-01976-w
Gorshkov, I.B., Maslova, N.V.: The group \(J_4\times J_4\) is recognizable by spectrum. J. Algebra Appl. 20(4), 2150061 (2021). https://doi.org/10.1142/S0219498821500614
Grechkoseeva, M.A.: Element orders in covers of finite simple groups of Lie type. J. Algebra Appl. 14(4), 1550056 (2015). https://doi.org/10.1142/S0219498815500565
Grechkoseeva, M.A., Vasil’ev, A.V.: On the structure of finite groups isospectral to finite simple groups. J. Group Theory 18(5), 741–759 (2015). https://doi.org/10.1515/jgth-2015-0019
Grechkoseeva, M.A., Mazurov, V.D., Shi, W., Vasil’ev, A.V., Yang, N.: Finite groups isospectral to simple groups. Commun. Math. Stat. (2022). https://doi.org/10.1007/s40304-022-00288-5
Isaacs, I.M.: Finite Group Theory. Graduate Studies in Mathematics, vol. 92. American Mathematical Society, Providence (2008)
Khukhro, E.I., Mazurov, V.D. (Eds.): Unsolved problems in group theory. The Kourovka notebook (2022). arXiv:1401.0300. https://kourovka-notebook.org
Mazurov, V.D.: Characterizations of finite groups by sets of orders of their elements. Algebra Log. 36(1), 23–32 (1997). https://doi.org/10.1007/BF02671951
Mazurov, V.D.: A characterization of finite nonsimple groups by the set of orders of their elements. Algebra Log. 36(3), 182–192 (1997). https://doi.org/10.1007/BF02671616
Mazurov, V.D.: Recognition of finite groups by a set of orders of their elements. Algebra Log. 37(6), 371–379 (1998). https://doi.org/10.1007/BF02671691
Robinson, D.J.S.: A Course in the Theory of Groups. Springer, New York (1996)
Vasil’ev, A.V.: On finite groups isospectral to simple classical groups. J. Algebra 423, 318–374 (2015). https://doi.org/10.1016/j.jalgebra.2014.10.013
Vasil’ev, A.V., Grechkoseeva, M.A.: On recognition by spectrum of finite simple linear groups over fields of characteristic \(2\). Siberian Math. J. 46(4), 593–600 (2005). https://doi.org/10.1007/s11202-005-0060-8
Vasil’ev, A.V., Vdovin, E.P.: An adjacency criterion for the prime graph of a finite simple group. Algebra Log. 44(6), 381–406 (2005). https://doi.org/10.1007/s10469-005-0037-5
Vasil’ev, A.V., Vdovin, E.P.: Cocliques of maximal size in the prime graph of a finite simple group. Algebra Log. 50(4), 291–322 (2011). https://doi.org/10.1007/s10469-011-9143-8
Vasil’ev, A.V., Grechkoseeva, M.A., Mazurov, V.D.: Characterization of the finite simple groups by spectrum and order. Algebra Log. 48(6), 385–409 (2009). https://doi.org/10.1007/s10469-009-9074-9
Wall, G.E.: On the conjugacy classes in the unitary, symplectic and orthogonal groups. J. Austral. Math. Soc. 3, 1–62 (1963). https://doi.org/10.1017/S1446788700027622
Wang, Zh., Vasil’ev, A.V., Grechkoseeva, M.A., Zhurtov, A.Kh.: A criterion for nonsolvability of a finite group and recognition of direct squares of simple groups. Algebra Log. (2023). https://doi.org/10.1007/s10469-023-09697-z
Yang, Y.: On analogues of Huppert’s conjecture. Bull. Aust. Math. Soc. 104(2), 272–277 (2021). https://doi.org/10.1017/S0004972720001409
Zavarnitsine, A.V., Mazurov, V.D.: On element orders in coverings of the simple groups \(L_n(q)\) and \(U_n(q)\). Proc. Steklov Inst. Math. 257(suppl. 1), S145–S154 (2007). https://doi.org/10.1134/S0081543807050100
Zhang, J.: Arithmetical conditions on element orders and group structure. Proc. Am. Math. Soc. 123(1), 39–44 (1995). https://doi.org/10.2307/2160607
Zsigmondy, K.: Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3, 265–284 (1892)
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The first and fourth authors were supported by Foreign Experts program in Jiangsu Province (No. JSB2018014). The second, third, and fourth authors were supported by the Program of Fundamental Research RAS, project FWNF-2022-0002.
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Yang, N., Gorshkov, I., Staroletov, A. et al. On recognition of direct powers of finite simple linear groups by spectrum. Annali di Matematica 202, 2699–2714 (2023). https://doi.org/10.1007/s10231-023-01336-9
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DOI: https://doi.org/10.1007/s10231-023-01336-9