On the composition factors of a group with the same prime graph as B n ( 5 )

Azam Babai; Behrooz Khosravi

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 2, page 469-486
  • ISSN: 0011-4642

Abstract

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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 p q . The prime graph of G is denoted by Γ ( 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 Γ ( G ) = Γ ( B n ( 5 ) ) , where n ≥ 6 , then G has a unique nonabelian composition factor isomorphic to B n ( 5 ) or C n ( 5 ) .

How to cite

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Babai, Azam, and Khosravi, Behrooz. "On the composition factors of a group with the same prime graph as $B_{n}(5)$." Czechoslovak Mathematical Journal 62.2 (2012): 469-486. <http://eudml.org/doc/246706>.

@article{Babai2012,
abstract = {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 $n\ge 6$, then $G$ has a unique nonabelian composition factor isomorphic to $B_\{n\}(5)$ or $C_\{n\}(5)$.},
author = {Babai, Azam, Khosravi, Behrooz},
journal = {Czechoslovak Mathematical Journal},
keywords = {prime graph; simple group; recognition; quasirecognition; prime graphs; finite simple groups; recognition; quasirecognition; sets of element orders},
language = {eng},
number = {2},
pages = {469-486},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the composition factors of a group with the same prime graph as $B_\{n\}(5)$},
url = {http://eudml.org/doc/246706},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Babai, Azam
AU - Khosravi, Behrooz
TI - On the composition factors of a group with the same prime graph as $B_{n}(5)$
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 2
SP - 469
EP - 486
AB - 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 $n\ge 6$, then $G$ has a unique nonabelian composition factor isomorphic to $B_{n}(5)$ or $C_{n}(5)$.
LA - eng
KW - prime graph; simple group; recognition; quasirecognition; prime graphs; finite simple groups; recognition; quasirecognition; sets of element orders
UR - http://eudml.org/doc/246706
ER -

References

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  1. Akhlaghi, Z., Khatami, M., Khosravi, B., 10.1007/s10474-009-8048-7, Acta Math. Hung. 122 (2009), 387-397. (2009) MR2481788DOI10.1007/s10474-009-8048-7
  2. Akhlaghi, Z., Khosravi, B., Khatami, M., 10.1142/S021819671000587X, Int. J. Algebra Comput. 20 (2010), 847-873. (2010) MR2738548DOI10.1142/S021819671000587X
  3. Babai, A., Khosravi, B., Hasani, N., Quasirecognition by prime graph of 2 D p ( 3 ) where p = 2 n + 1 ≥ 5 is a prime, Bull. Malays. Math. Sci. Soc. 32 (2009), 343-350. (2009) MR2562173
  4. Babai, A., Khosravi, B., 10.1134/S003744661105003X, Sib. Math. J. 52 (2011), 993-1003. (2011) MR2908121DOI10.1134/S003744661105003X
  5. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A., Atlas of Finite Groups, Clarendon Press Oxford (1985). (1985) Zbl0568.20001MR0827219
  6. Guralnick, R. M., Tiep, P. H., 10.1515/jgth.2003.020, J. Group Theory 6 (2003), 271-310. (2003) Zbl1046.20013MR1983368DOI10.1515/jgth.2003.020
  7. Hagie, M., 10.1081/AGB-120022800, Comm. Algebra 31 (2003), 4405-4424. (2003) Zbl1031.20009MR1995543DOI10.1081/AGB-120022800
  8. He, H., Shi, W., 10.1142/S0218196709005299, Int. J. Algebra Comput. 19 (2009), 681-698. (2009) Zbl1182.20016MR2547064DOI10.1142/S0218196709005299
  9. Khatami, M., Khosravi, B., Akhlaghi, Z., NCF-distinguishability by prime graph of P G L ( 2 , p ) , where p is a prime, Rocky Mt. J. Math. 41 (2011), 1523-1545. (2011) MR2838076
  10. Khosravi, B., Babai, A., 10.1007/s00605-009-0155-6, Monatsh. Math. 162 (2011), 289-296. (2011) MR2775847DOI10.1007/s00605-009-0155-6
  11. Khosravi, A., Khosravi, B., 10.1007/s11202-007-0059-4, Sib. Math. J. 48 (2007), 570-577. (2007) MR2347918DOI10.1007/s11202-007-0059-4
  12. Khosravi, B., Khosravi, A., 10.1007/s11202-008-0072-2, Sib. Math. J. 49 (2008), 749-757. (2008) MR2456703DOI10.1007/s11202-008-0072-2
  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. (2007) Zbl1133.20008MR2350073
  14. Khosravi, B., Khosravi, B., Khosravi, B., 10.1007/s10474-007-6021-x, Acta Math. Hung. 116 (2007), 295-307. (2007) MR2335801DOI10.1007/s10474-007-6021-x
  15. Khosravi, B., Khosravi, B., Khosravi, B., 10.1007/s00229-007-0160-9, Manuscr. Math. 126 (2008), 49-58. (2008) Zbl1143.20009MR2395248DOI10.1007/s00229-007-0160-9
  16. Khosravi, B., 10.1007/s11202-009-0040-5, Sib. Math. J. 50 (2009), 355-359. (2009) Zbl1212.20047MR2531768DOI10.1007/s11202-009-0040-5
  17. Khosravi, B., Some characterizations of L 9 ( 2 ) related to its prime graph, Publ. Math. 75 (2009), 375-385. (2009) Zbl1207.20008MR2588212
  18. Khosravi, B., 10.1142/S0219498808003090, J. Algebra Appl. 7 (2008), 735-748. (2008) MR2483329DOI10.1142/S0219498808003090
  19. Khosravi, B., Moradi, H., 10.1007/s10474-010-0053-3, Acta. Math. Hung. 132 (2011), 140-153. (2011) Zbl1232.20020MR2805484DOI10.1007/s10474-010-0053-3
  20. Mazurov, V. D., 10.1007/BF02671951, Algebra Logic 36 (1997), 23-32. (1997) MR1454690DOI10.1007/BF02671951
  21. Sierpiński, W., Elementary Theory of Numbers (Monografie Matematyczne Vol. 42), Państwowe Wydawnictwo Naukowe Warsaw (1964). (1964) MR0175840
  22. Stensholt, E., 10.1016/0021-8693(78)90211-9, J. Algebra 53 (1978), 136-187. (1978) Zbl0386.20006MR0486182DOI10.1016/0021-8693(78)90211-9
  23. Vasil'ev, A. V., Vdovin, E. P., 10.1007/s10469-005-0037-5, Algebra Logic 44 (2005), 381-405. (2005) MR2213302DOI10.1007/s10469-005-0037-5
  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 2893582. MR2893582
  25. Vasil'ev, A. V., Gorshkov, I. B., 10.1007/s11202-009-0027-2, Sib. Math. J. 50 (2009), 233-238. (2009) MR2531755DOI10.1007/s11202-009-0027-2
  26. Vasil'ev, A. V., Grechkoseeva, M. A., 10.1023/B:SIMJ.0000028607.23176.5f, Sib. Math. J. 45 (2004), 420-431. (2004) MR2078712DOI10.1023/B:SIMJ.0000028607.23176.5f
  27. Zavarnitsin, A. V., 10.1007/s10469-006-0020-9, Algebra Logic 43 (2006), 220-231. (2006) MR2287647DOI10.1007/s10469-006-0020-9
  28. Zsigmondy, K., 10.1007/BF01692444, Monatsh. Math. Phys. 3 (1892), 265-284 German. (1892) MR1546236DOI10.1007/BF01692444

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