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Groups with the same prime graph as the simple group \(D_n(5)\). (English) Zbl 1311.20012

Ukr. Math. J. 66, No. 5, 666-677 (2014) and Ukr. Mat. Zh. 66, No. 5, 598-608 (2014).
From the text: A finite group \(G\) is called recognizable by the prime graph if \(\Gamma(H)=\Gamma(G)\) implies that \(H\cong G\). A non-Abelian simple group \(P\) is called quasirecognizable by the prime graph if every finite group whose prime graph is \(\Gamma(P)\) has a unique non-Abelian composition factor isomorphic to \(P\). It is clear that the recognition (quasirecognition) by the prime graph implies the recognition (quasirecognition) by the spectrum but the converse is in general not true. Moreover, some methods used for the recognition by the spectrum cannot be used for the recognition by the prime graph.
Let \(G\) be a finite group such that \(\Gamma(G)=\Gamma(D_n(5))\), where \(n\geq 6\). In the present paper, as the main result, we show that if \(n\) is odd, then \(G\) is recognizable by the prime graph and if \(n\) is even, then \(G\) is quasirecognizable by the prime graph.

MSC:

20D06 Simple groups: alternating groups and groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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[1] Z. Akhlaghi, M. Khatami, and B. Khosravi, “Quasirecognition by prime graph of the simple group 2<Emphasis Type=”Italic“>F4(<Emphasis Type=”Italic“>q),” Acta Math. Hung., 122, No. 4, 387-397 (2009). · Zbl 1181.20012 · doi:10.1007/s10474-009-8048-7
[2] Z. Akhlaghi, M. Khatami, and B. Khosravi, “Characterization by prime graph of <Emphasis Type=”Italic“>PGL(2<Emphasis Type=”Italic“>,pk)<Emphasis Type=”Italic“>, where <Emphasis Type=”Italic“>p and <Emphasis Type=”Italic“>k >1 are odd,” Int. J. Algebra Comput., 20, No. 7, 847-873 (2010). · Zbl 1216.20006 · doi:10.1142/S021819671000587X
[3] A. Babai, B. Khosravi, and N. Hasani, “Quasirecognition by prime graph of 2<Emphasis Type=”Italic“>Dp(3) where <Emphasis Type=”Italic“>p = 2<Emphasis Type=”Italic“>n +1 ≥ 5 is a prime,” Bull. Malays. Math. Sci. Soc., 32, No. 3, 343-350 (2009). · Zbl 1172.20015
[4] A. Babai and B. Khosravi, “Recognition by prime graph of 2<Emphasis Type=”Italic“>D2<Emphasis Type=”Italic“>m+1(3),” Sib. Math. J., 52, No. 5, 993-1003 (2011). · Zbl 1237.20014 · doi:10.1134/S003744661105003X
[5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Oxford Univ. Press, Oxford (1985). · Zbl 0568.20001
[6] M. A. Grechkoseeva,W. J. Shi, and A.V. Vasil’ev, “Recognition by spectrum of <Emphasis Type=”Italic“>L16(2<Emphasis Type=”Italic“>m),” Alg. Colloq., 14, No. 3, 462-470 (2007).
[7] R. M. Guralnick and P. H. Tiep, “Finite simple unisingular groups of Lie type,” J. Group Theory, 6, 271-310 (2003). · Zbl 1046.20013 · doi:10.1515/jgth.2003.020
[8] M. Hagie, “The prime graph of a sporadic simple group,” Comm. Algebra, 31, No. 9, 4405-4424 (2003). · Zbl 1031.20009 · doi:10.1081/AGB-120022800
[9] H. He and W. Shi, “Recognition of some finite simple groups of type <Emphasis Type=”Italic“>Dn(<Emphasis Type=”Italic“>q) by spectrum,” Int. J. Algebra Comput., 19, No. 5, 681-698 (2009). · Zbl 1182.20016 · doi:10.1142/S0218196709005299
[10] M. Khatami, B. Khosravi, and Z. Akhlaghi, “NCF-distinguishability by prime graph of <Emphasis Type=”Italic“>PGL(2<Emphasis Type=”Italic“>,p)<Emphasis Type=”Italic“>, where <Emphasis Type=”Italic“>p is a prime,” Rocky Mountain J. Math., 41, No. 5, 1523-1545 (2011). · Zbl 1246.20032 · doi:10.1216/RMJ-2011-41-5-1523
[11] A. Khosravi and B. Khosravi, “Quasirecognition by prime graph of the simple group 2<Emphasis Type=”Italic“>G2(<Emphasis Type=”Italic“>q),” Sib. Math. J., 48, No. 3, 570-577 (2007). · doi:10.1007/s11202-007-0059-4
[12] B. Khosravi and A. Babai, “Quasirecognition by prime graph of <Emphasis Type=”Italic“>F4(<Emphasis Type=”Italic“>q) where <Emphasis Type=”Italic“>q =2<Emphasis Type=”Italic“>n >2,” Monatsh. Math., 162, No. 3, 289-296 (2011). · Zbl 1216.20007 · doi:10.1007/s00605-009-0155-6
[13] B. Khosravi, B. Khosravi, and B. Khosravi, “2-Recognizability of <Emphasis Type=”Italic“>PSL(2<Emphasis Type=”Italic“>,p2) by the prime graph,” Sib. Math. J., 49, No. 4, 749-757 (2008). · Zbl 1154.20007 · doi:10.1007/s11202-008-0072-2
[14] B. Khosravi, B. Khosravi, and B. Khosravi, “Groups with the same prime graph as a CIT simple group,” Houston J. Math., 33, No. 4, 967-977 (2007). · Zbl 1133.20008
[15] B. Khosravi, B. Khosravi, and B. Khosravi, “On the prime graph of <Emphasis Type=”Italic“>PSL(2<Emphasis Type=”Italic“>,p) where <Emphasis Type=”Italic“>p >3 is a prime number,” Acta Math. Hung., 116, No. 4, 295-307 (2007). · Zbl 1149.20015 · doi:10.1007/s10474-007-6021-x
[16] B. Khosravi, B. Khosravi, and B. Khosravi, “A characterization of the finite simple group <Emphasis Type=”Italic“>L16(2) by its prime graph,” Manuscr. Math., 126, 49-58 (2008). · Zbl 1143.20009 · doi:10.1007/s00229-007-0160-9
[17] B. Khosravi, “Quasirecognition by prime graph of <Emphasis Type=”Italic“>L10(2),” Sib. Math. J., 50, No. 2, 355-359 (2009). · doi:10.1007/s11202-009-0040-5
[18] B. Khosravi, “Some characterizations of <Emphasis Type=”Italic“>L9(2) related to its prime graph,” Publ. Math. Debrecen, 75, No. 3-4, 375-385 (2009). · Zbl 1207.20008
[19] B. Khosravi, “<Emphasis Type=”Italic“>n-Recognition by prime graph of the simple group <Emphasis Type=”Italic“>PSL(2<Emphasis Type=”Italic“>, q),” J. Algebra Appl., 7, No. 6, 735-748 (2008). · Zbl 1177.20023 · doi:10.1142/S0219498808003090
[20] B. Khosravi and H. Moradi, “Quasirecognition by prime graph of finite simple groups <Emphasis Type=”Italic“>Ln(2) and <Emphasis Type=”Italic“>Un(2),” Acta. Math. Hung., 132, No. 12, 140-153 (2011). · Zbl 1232.20020 · doi:10.1007/s10474-010-0053-3
[21] M. S. Lucido, “Prime graph components of finite almost simple groups,” Rend. Semin. Mat. Univ. Padova, 102, 1-14 (1999). · Zbl 0941.20008
[22] V. D. Mazurov, “Characterizations of finite groups by the set of orders of their elements,” Alg. Logik., 36, No. 1, 23-32 (1997). · doi:10.1007/BF02671951
[23] V. D. Mazurov and G. Y. Chen, “Recognizability of finite simple groups <Emphasis Type=”Italic“>L4(2<Emphasis Type=”Italic“>m) and <Emphasis Type=”Italic“>U4(2<Emphasis Type=”Italic“>m) by the spectrum,” Alg. Logik., 47, No. 1, 83-93 (2008). · Zbl 1155.20023
[24] W. Sierpi´nski, Elementary Theory of Numbers, PWN, Warsaw (1964), Vol. 42. · Zbl 0638.10001
[25] E. Stensholt, “Certain embeddings among finite groups of Lie type,” J. Algebra, 53, 136-187 (1978). · Zbl 0386.20006 · doi:10.1016/0021-8693(78)90211-9
[26] A.V. Vasil’ev and E. P. Vdovin, “Adjacency criterion in the prime graph of a finite simple group,” Alg. Logik., 44, No. 6, 381-405 (2005). · Zbl 1104.20018 · doi:10.1007/s10469-005-0037-5
[27] A. V. Vasil’ev and E. P. Vdovin, “Cocliques of maximal size in the prime graph of a finite simple group,” http://arxiv.org/abs/0905.1164v1. · Zbl 1256.05105
[28] A.V. Vasil’ev and I. B. Gorshkov, “On the recognition of finite simple groups with connected prime graph,” Sib. Math. Zh., 50, No. 2, 233-238 (2009). · doi:10.1007/s11202-009-0027-2
[29] A.V. Vasil’ev and M. A. Grechkoseeva, “On the recognition by spectrum of finite simple linear groups over the fields of characteristic 2,” Sib. Mat. Zh., 46, No. 4, 749-758 (2005). · Zbl 1117.20019
[30] A.V. Vasil’ev and M. A. Grechkoseeva, “On the recognition of finite simple orthogonal groups of dimension 2<Emphasis Type=”Italic“>m, 2<Emphasis Type=”Italic“>m+1 and 2<Emphasis Type=”Italic“>m+2 over the field of characteristic 2,” Sib. Math. Zh., 45, No. 3, 420-431 (2004). · doi:10.1023/B:SIMJ.0000028607.23176.5f
[31] A.V. Vasil’ev, M. A. Grechkoseeva, and V. D. Mazurov, “Characterization of finite simple groups by the spectrum and order,” Alg. Logik., 48, No. 6, 385-409 (2009). · Zbl 1245.20012 · doi:10.1007/s10469-009-9074-9
[32] A.V. Zavarnitsin, “On the recognition of finite groups by the prime graph,” Alg. Logik., 43, No. 4, 220-231 (2006). · Zbl 1115.20011 · doi:10.1007/s10469-006-0020-9
[33] K. Zsigmondy, “Zur theorie der potenzreste,” Monatsh. Math. Phys., 3, 265-284 (1892). · JFM 24.0176.02 · doi:10.1007/BF01692444
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