# Recognition by spectrum for finite simple linear groups of small dimensions over fields of characteristic 2

Article

First Online:

Received:

Two groups are said to be isospectral if they share the same set of element orders. For every finite simple linear group L of dimension n over an arbitrary field of characteristic 2, we prove that any finite group G isospectral to L is isomorphic to an automorphic extension of L. An explicit formula is derived for the number of isomorphism classes of finite groups that are isospectral to L. This account is a continuation of the second author's previous paper where a similar result was established for finite simple linear groups L in a sufficiently large dimension (n > 26), and so here we confine ourselves to groups of dimension at most 26.

### Keywords

finite simple group linear group order of element spectrum of group recognition by spectrum## Preview

Unable to display preview. Download preview PDF.

### References

- 1.V. D. Mazurov, “Groups with prescribed spectrum,”
*Izv. Ural. Gos. Univ., Mat. Mekh.*, Issue 7, No. 36, 119–138 (2005).Google Scholar - 2.M. A. Grechkoseeva, W. J. Shi, and A. V. Vasil'ev, “Recognition by spectrum of finite simple groups of Lie type,”
*Front. Math. China*,**3**, No. 2, 275–285 (2008).MATHCrossRefMathSciNetGoogle Scholar - 3.W. Shi, “A characteristic property of
*J*_{1}and*PSL*_{2}(2^{n})” [in Chinese],*Adv. Math.*,**16**, 397–401 (1987).MATHGoogle Scholar - 4.F. J. Liu, “A characteristic property of projective special linear group
*L*_{3}(8)” [in Chinese],*J. Southwest-China Normal Univ.*,**22**, No. 2, 131–134 (1997).Google Scholar - 5.V. D. Mazurov, M. C. Xu, and H. P. Cao, “Recognition of finite simple groups
*L*_{3}(2^{m}) and*U*_{3}(2^{m}) by their element orders,”*Algebra Logika*,**39**, No. 5, 567–585 (2000).MATHGoogle Scholar - 6.V. D. Mazurov and G. Y. Chen, “Recognizability of finite simple groups
*L*_{4}(2^{m}) and*U*_{4}(2^{m}) by spectrum,”*Algebra Logika*,**47**, No. 1, 83–93 (2008).MathSciNetGoogle Scholar - 7.M. A. Grechkoseeva, W. J. Shi, and A. V. Vasil'ev, “Recognition by spectrum of
*L*_{16}(2^{m}),”*Alg. Colloq.*,**14**, No. 4, 585–591 (2007).MATHMathSciNetGoogle Scholar - 8.M. A. Grechkoseeva, “Recognition by spectrum for finite linear groups over fields of characteristic 2,”
*Algebra Logika*,**47**, No. 4, 405–427 (2008).Google Scholar - 9.A. V. Vasil'ev and M. A. Grechkoseeva, “On recognition by spectrum of finite simple linear groups over fields of characteristic 2,”
*Sib. Mat. Zh.*,**46**, No. 4, 749–758 (2005).MATHMathSciNetGoogle Scholar - 10.M. A. Grechkoseeva, M. S. Lucido, V. D. Mazurov, A. R. Moghaddamfar, and A. V. Vasil'ev, “On recognition of the projective special linear groups over the binary field,”
*Sib. Electr. Math. Rep.*,**2**, 253–263 (2005); http://semr.math.nsc.ru.MATHMathSciNetGoogle Scholar - 11.A. V. Zavarnitsine and V. D. Mazurov, “Orders of elements in coverings of finite simple linear and unitary groups and recognizability of
*L*_{n}(2) by spectrum,”*Dokl. Akad. Nauk*,**409**, No. 6, 736–739 (2006).MathSciNetGoogle Scholar - 12.A. V. Zavarnitsine, “Properties of element orders in covers for
*L*_{n}(*q*) and*U*_{n}(*q*),”*Sib. Mat. Zh.*,**49**, No. 2, 309–322 (2008).Google Scholar - 13.A. V. Vasil'ev, “On connection between the structure of a finite group and properties of its prime graph,”
*Sib. Mat. Zh.*,**46**, No. 3, 511–522 (2005).MATHGoogle Scholar - 14.A. V. Vasilyev and I. B. Gorshkov, “On recognition of finite simple groups with connected prime graph,”
*Sib. Mat. Zh.*, to appear.Google Scholar - 15.V. D. Mazurov, “Characterizations of finite groups by sets of orders of their elements,”
*Algebra Logika*,**36**, No. 1, 37–53 (1997).MATHMathSciNetGoogle Scholar - 16.A. A. Buturlakin, “Spectra of finite linear and unitary groups,”
*Algebra Logika*,**47**, No. 2, 157–173 (2008).MathSciNetGoogle Scholar - 17.K. Zsigmondy, “Zur Theorie der Potenzreste,”
*Monatsh. Math. Phys.*,**3**, 265–284 (1892).CrossRefMathSciNetGoogle Scholar - 18.B. Huppert and N. Blackburn,
*Finite Groups. II*,*Grundlehren Math. Wiss.*,**242**, Springer-Verlag, Berlin (1982).MATHGoogle Scholar - 19.A. V. Vasiliev and E. P. Vdovin, “An adjacency criterion for the prime graph of a finite simple group,”
*Algebra Logika*,**44**, No. 6, 682–725 (2005).MATHMathSciNetGoogle Scholar

## Copyright information

© Springer Science+Business Media, Inc. 2008