On almost recognizability by spectrum of simple classical groups

Document Type: Research Paper

Author

Abstract

‎The set of element orders of a finite group $G$ is called the {\em spectrum}‎. ‎Groups with coinciding spectra are said to be {\em isospectral}‎. ‎It is known that if $G$ has a nontrivial normal soluble subgroup then there exist infinitely many pairwise non-isomorphic‎ ‎groups isospectral to $G$‎. ‎The situation is quite different if $G$ is a nonabelain simple group‎. ‎Recently it was proved that if $L$ is a simple classical group of dimension at least 62 and $G$ is a finite group‎ ‎isospectral to $L$‎, ‎then up to isomorphism $L\leq G\leq\Aut L$‎. ‎We show that the assertion remains true‎ ‎if 62 is replaced by 38‎.

Keywords

Main Subjects


[1] A. A. Buturlakin, Sp ectra of nite linear and unitary groups, Algebra Logic, 47 no. 2 (2008) 91-99.

[2] A. A. Buturlakin, Sp ectra of nite symplectic and orthogonal groups, Siberian Adv. Math., 21 no. 3 (2011) 176-210.

[3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of nite groups, Clarendon Press, Oxford, 1985.

[4] P. Erdos, On the co efficients of the cyclotomic p olynomial, Bull. Amer. Math. Soc., 52 (1946) 179-184.

[5] M. A. Grechkoseeva, On element orders in covers of nite simple groups of Lie type, J. Algebra Appl, 14 (2015) 16 pages.

[6] M. A. Grechkoseeva and A. V. Vasil ′ ev, On the structure of nite groups isosp ectral to nite simple groups, J. Group Theory, 18 no. 5 (2015) 741-759.

[7] A. S. Kondrat ′ ev, On prime graph comp onents of nite simple groups, Math. USSR-Sb., 67 no. 1 (1990) 235-247.

[8] A. S. Kondrat ′ ev and V. D. Mazurov, Recognition of alternating groups of prime degree from the orders of their elements, Siberian Math. J., 41 no. 2 (2000) 294-302.

[9] V. D. Mazurov, Recognition of nite groups by a set of orders of their elements, Algebra Logic, 37 no. 6 (1998) 371-379.

[10] M. Roitman, On Zsigmondy primes, Proc. Amer. Math. Soc., 125 no. 7 (1997) 1913-1919.

[11] W. J. Shi, The characterization of the sp oradic simple groups by their element orders, Algebra Col loq., 1 no. 2 (1994) 156-166.

[12] A. V. Vasil ′ ev, On nite groups isosp ectral to simple classical groups, J. Algebra, 423 (2015) 318-374.

[13] A. V. Vasil ′ ev, On connection b etween the structure of a nite group and the prop erties of its prime graph, Siberian Math. J., 46 no. 3 (2005) 396-404.

[14] A. V. Vasil ′ ev and I. B. Gorshkov, On recognition of nite simple groups with connected prime graph, Siberian Math. J., 50 no. 2 (2009) 233-238.

[15] A. V. Vasil′ev, I. B. Gorshkov, M. A. Grechkoseeva, A. S. Kondrat′ev and A. M. Staroletov, On recognizability by sp ectrum of nite simple groups of typ es Bn, Cn, and 2Dn for n=2k , Proc. Steklov Inst. Math., 267 suppl 1 (2009) 218-233.

[16] A. V. Vasil ′ ev and M. A. Grechkoseeva, Recognition by sp ectrum for simple classical groups in characteristic 2, Siberian Math. J., 56 no. 6 (2015) 1009-1018.

[17] A. V. Vasil ′ ev, M. A. Grechkoseeva and V. D. Mazurov, Characterization of the nite simple groups by sp ectrum and order, Algebra Logic, 48 no. 6 (2009) 385-409.

[18] A. V. Vasil ′ ev, M. A. Grechkoseeva and V. D. Mazurov, On nite groups isosp ectral to simple symplectic and orthogonal groups, Siberian Math. J., 50 no. 6 (2009) 965-981.

[19] A. V. Vasil ′ ev, M. A. Grechkoseeva and A. M. Staroletov, On nite groups isosp ectral to simple linear and unitary groups, Siberian Math. J., 52 no. 1 (2011) 30-40.

[20] A. V. Vasil ′ ev and E. P. Vdovin, An adjacency criterion for the prime graph of a nite simple group, Algebra Logic, 44 no. 6 (2005) 381-406.

[21] A. V. Vasil ′ ev and E. P. Vdovin, Co cliques of maximal size in the prime graph of a nite simple group, Algebra Logic, 50 no. 4 (2011) 291-322.

[22] J. S. Williams, Prime graph comp onents of nite groups, J. Algebra, 69 (1981) 487-513.

[23] A. V. Zavarnitsine, Prop erties of element orders in covers for Ln(q) and Un(q), Siberian Math. J., 49 no. 2 (2008) 246-256.

[24] K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys., 3 (1892) 265-284.