Previous |  Up |  Next

Article

Title: On the composition factors of a group with the same prime graph as $B_{n}(5)$ (English)
Author: Babai, Azam
Author: Khosravi, Behrooz
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 62
Issue: 2
Year: 2012
Pages: 469-486
Summary lang: English
.
Category: math
.
Summary: Let $G$ be a finite group. The prime graph of $G$ is a graph whose vertex set is the set of prime divisors of $|G|$ and two distinct primes $p$ and $q$ are joined by an edge, whenever $G$ contains an element of order $pq$. The prime graph of $G$ is denoted by $\Gamma (G)$. It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if $G$ is a finite group such that $\Gamma (G)=\Gamma (B_{n}(5))$, where $n\geq 6$, then $G$ has a unique nonabelian composition factor isomorphic to $B_{n}(5)$ or $C_{n}(5)$. (English)
Keyword: prime graph
Keyword: simple group
Keyword: recognition
Keyword: quasirecognition
MSC: 05C25
MSC: 20D05
MSC: 20D06
MSC: 20D60
idZBL: Zbl 1249.20014
idMR: MR2990187
DOI: 10.1007/s10587-012-0022-9
.
Date available: 2012-06-08T09:46:43Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/142839
.
Reference: [1] Akhlaghi, Z., Khatami, M., Khosravi, B.: Quasirecognition by prime graph of the simple group $^2F_4(q)$.Acta Math. Hung. 122 (2009), 387-397. MR 2481788, 10.1007/s10474-009-8048-7
Reference: [2] Akhlaghi, Z., Khosravi, B., Khatami, M.: Characterization by prime graph of $ PGL(2,p^k)$ where $p$ and $k>1$ are odd.Int. J. Algebra Comput. 20 (2010), 847-873. MR 2738548, 10.1142/S021819671000587X
Reference: [3] Babai, A., Khosravi, B., Hasani, N.: Quasirecognition by prime graph of $^2D_p(3)$ where $p=2^n+1\geq5$ is a prime.Bull. Malays. Math. Sci. Soc. 32 (2009), 343-350. MR 2562173
Reference: [4] Babai, A., Khosravi, B.: Recognition by prime graph of $^2D_{2^m+1}(3)$.Sib. Math. J. 52 (2011), 993-1003. MR 2908121, 10.1134/S003744661105003X
Reference: [5] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A.: Atlas of Finite Groups.Clarendon Press Oxford (1985). Zbl 0568.20001, MR 0827219
Reference: [6] Guralnick, R. M., Tiep, P. H.: Finite simple unisingular groups of Lie type.J. Group Theory 6 (2003), 271-310. Zbl 1046.20013, MR 1983368, 10.1515/jgth.2003.020
Reference: [7] Hagie, M.: The prime graph of a sporadic simple group.Comm. Algebra 31 (2003), 4405-4424. Zbl 1031.20009, MR 1995543, 10.1081/AGB-120022800
Reference: [8] He, H., Shi, W.: Recognition of some finite simple groups of type $D_n(q)$ by spectrum.Int. J. Algebra Comput. 19 (2009), 681-698. Zbl 1182.20016, MR 2547064, 10.1142/S0218196709005299
Reference: [9] Khatami, M., Khosravi, B., Akhlaghi, Z.: NCF-distinguishability by prime graph of $PGL(2,p)$, where $p$ is a prime.Rocky Mt. J. Math. 41 (2011), 1523-1545. MR 2838076
Reference: [10] Khosravi, B., Babai, A.: Quasirecognition by prime graph of $F_{4}(q)$ where $q=2^n>2$.Monatsh. Math. 162 (2011), 289-296. MR 2775847, 10.1007/s00605-009-0155-6
Reference: [11] Khosravi, A., Khosravi, B.: Quasirecognition by prime graph of the simple group $^2G_{2}(q)$.Sib. Math. J. 48 (2007), 570-577. MR 2347918, 10.1007/s11202-007-0059-4
Reference: [12] Khosravi, B., Khosravi, A.: $2$-recognizability by the prime graph of $ PSL(2,p^2)$.Sib. Math. J. 49 (2008), 749-757. MR 2456703, 10.1007/s11202-008-0072-2
Reference: [13] Khosravi, B., Khosravi, B., Khosravi, B.: Groups with the same prime graph as a CIT simple group.Houston J. Math. 33 (2007), 967-977. Zbl 1133.20008, MR 2350073
Reference: [14] Khosravi, B., Khosravi, B., Khosravi, B.: On the prime graph of $ PSL(2, p)$ where $p>3$ is a prime number.Acta Math. Hung. 116 (2007), 295-307. MR 2335801, 10.1007/s10474-007-6021-x
Reference: [15] Khosravi, B., Khosravi, B., Khosravi, B.: A characterization of the finite simple group $L_{16}(2)$ by its prime graph.Manuscr. Math. 126 (2008), 49-58. Zbl 1143.20009, MR 2395248, 10.1007/s00229-007-0160-9
Reference: [16] Khosravi, B.: Quasirecognition by prime graph of $L_{10}(2)$.Sib. Math. J. 50 (2009), 355-359. Zbl 1212.20047, MR 2531768, 10.1007/s11202-009-0040-5
Reference: [17] Khosravi, B.: Some characterizations of $L_{9}(2)$ related to its prime graph.Publ. Math. 75 (2009), 375-385. Zbl 1207.20008, MR 2588212
Reference: [18] Khosravi, B.: $n$-recognition by prime graph of the simple group $ PSL(2,q)$.J. Algebra Appl. 7 (2008), 735-748. MR 2483329, 10.1142/S0219498808003090
Reference: [19] Khosravi, B., Moradi, H.: Quasirecognition by prime graph of finite simple groups $L_n(2)$ and $U_n(2)$.Acta. Math. Hung. 132 (2011), 140-153. Zbl 1232.20020, MR 2805484, 10.1007/s10474-010-0053-3
Reference: [20] Mazurov, V. D.: Characterizations of finite groups by the set of orders of their elements.Algebra Logic 36 (1997), 23-32. MR 1454690, 10.1007/BF02671951
Reference: [21] Sierpiński, W.: Elementary Theory of Numbers (Monografie Matematyczne Vol. 42).Państwowe Wydawnictwo Naukowe Warsaw (1964). MR 0175840
Reference: [22] Stensholt, E.: Certain embeddings among finite groups of Lie type.J. Algebra 53 (1978), 136-187. Zbl 0386.20006, MR 0486182, 10.1016/0021-8693(78)90211-9
Reference: [23] Vasil'ev, A. V., Vdovin, E. P.: An adjacency criterion in the prime graph of a finite simple group.Algebra Logic 44 (2005), 381-405. MR 2213302, 10.1007/s10469-005-0037-5
Reference: [24] Vasil'ev, A. V., Vdovin, E. P.: Cocliques of maximal size in the prime graph of a finite simple group.http://arxiv.org/abs/0905.1164v1 \MR 2893582. MR 2893582
Reference: [25] Vasil'ev, A. V., Gorshkov, I. B.: On the recognition of finite simple groups with a connected prime graph.Sib. Math. J. 50 (2009), 233-238. MR 2531755, 10.1007/s11202-009-0027-2
Reference: [26] Vasil'ev, A. V., Grechkoseeva, M. A.: On recognition of the finite simple orthogonal groups of dimension $2^m$, $2^m+1$, and $2^m+2$ over a field of characteristic 2.Sib. Math. J. 45 (2004), 420-431. MR 2078712, 10.1023/B:SIMJ.0000028607.23176.5f
Reference: [27] Zavarnitsin, A. V.: On the recognition of finite groups by the prime graph.Algebra Logic 43 (2006), 220-231. MR 2287647, 10.1007/s10469-006-0020-9
Reference: [28] Zsigmondy, K.: Zur Theorie der Potenzreste.Monatsh. Math. Phys. 3 (1892), 265-284 German. MR 1546236, 10.1007/BF01692444
.

Files

Files Size Format View
CzechMathJ_62-2012-2_10.pdf 293.9Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo