Abstract
For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set with a maximal number of vertices and an independent set with a maximal number of vertices containing 2, and to define orders on these sets; the information obtained is collected in tables. We consider several applications of these results to various problems in finite group theory, in particular, to the recognition-by-spectra problem for finite groups.
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Additional information
Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1; by the RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 8294; by FP “Universities of Russia,” grant No. UR.04.01.202; and by Presidium SB RAS grant No. 86-197.
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Translated from Algebra i Logika, Vol. 44, No. 6, pp. 682–725, November–December, 2005.
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Vasiliev, A.V., Vdovin, E.P. An Adjacency Criterion for the Prime Graph of a Finite Simple Group. Algebr Logic 44, 381–406 (2005). https://doi.org/10.1007/s10469-005-0037-5
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DOI: https://doi.org/10.1007/s10469-005-0037-5