# Groups with the Same Prime Graph as the Simple Group *D* _{ n }(5)

Article

First Online:

- 34 Downloads

Let *G* be a finite group. The prime graph of *G* is denoted by Γ(*G*). Let *G* be a finite group such that Γ(*G*) = Γ(*D* _{ n }(5)), where *n* ≥ 6. In the 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.

## Keywords

Prime Number Simple Group Prime Divisor Prime Graph Sporadic Simple Group
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Z. Akhlaghi, M. Khatami, and B. Khosravi, “Quasirecognition by prime graph of the simple group 2
*F*4(*q*),”*Acta Math. Hung.*, 122, No. 4, 387–397 (2009).CrossRefzbMATHMathSciNetGoogle Scholar - 2.Z. Akhlaghi, M. Khatami, and B. Khosravi, “Characterization by prime graph of
*PGL*(2*,pk*)*,*where*p*and*k >*1 are odd,”*Int. J. Algebra Comput.*, 20, No. 7, 847–873 (2010).CrossRefzbMATHMathSciNetGoogle Scholar - 3.A. Babai, B. Khosravi, and N. Hasani, “Quasirecognition by prime graph of 2
*Dp*(3) where*p*= 2*n*+1 ≥ 5 is a prime,”*Bull. Malays. Math. Sci. Soc.*, 32, No. 3, 343–350 (2009).zbMATHMathSciNetGoogle Scholar - 4.A. Babai and B. Khosravi, “Recognition by prime graph of 2
*D*2*m*+1(3),”*Sib. Math. J.*, 52, No. 5, 993–1003 (2011).CrossRefMathSciNetGoogle Scholar - 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).zbMATHGoogle Scholar - 6.M. A. Grechkoseeva,W. J. Shi, and A.V. Vasil’ev, “Recognition by spectrum of
*L*16(2*m*),”*Alg. Colloq.*, 14, No. 3, 462–470 (2007).MathSciNetGoogle Scholar - 7.R. M. Guralnick and P. H. Tiep, “Finite simple unisingular groups of Lie type,”
*J. Group Theory*, 6, 271–310 (2003).CrossRefzbMATHMathSciNetGoogle Scholar - 8.M. Hagie, “The prime graph of a sporadic simple group,”
*Comm. Algebra*, 31, No. 9, 4405–4424 (2003).CrossRefzbMATHMathSciNetGoogle Scholar - 9.H. He and W. Shi, “Recognition of some finite simple groups of type
*Dn*(*q*) by spectrum,”*Int. J. Algebra Comput.*, 19, No. 5, 681–698 (2009).CrossRefzbMATHMathSciNetGoogle Scholar - 10.M. Khatami, B. Khosravi, and Z. Akhlaghi, “NCF-distinguishability by prime graph of
*PGL*(2*,p*)*,*where*p*is a prime,”*Rocky Mountain J. Math.*, 41, No. 5, 1523–1545 (2011).CrossRefzbMATHMathSciNetGoogle Scholar - 11.A. Khosravi and B. Khosravi, “Quasirecognition by prime graph of the simple group 2
*G*2(*q*),”*Sib. Math. J.*, 48, No. 3, 570–577 (2007).CrossRefMathSciNetGoogle Scholar - 12.B. Khosravi and A. Babai, “Quasirecognition by prime graph of
*F*4(*q*) where*q*=2*n >*2,”*Monatsh. Math.*, 162, No. 3, 289–296 (2011).CrossRefzbMATHMathSciNetGoogle Scholar - 13.B. Khosravi, B. Khosravi, and B. Khosravi, “2-Recognizability of
*PSL*(2*,p*2) by the prime graph,”*Sib. Math. J.*, 49, No. 4, 749–757 (2008).CrossRefMathSciNetGoogle Scholar - 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).zbMATHMathSciNetGoogle Scholar - 15.B. Khosravi, B. Khosravi, and B. Khosravi, “On the prime graph of
*PSL*(2*,p*) where*p >*3 is a prime number,”*Acta Math. Hung.*, 116, No. 4, 295–307 (2007).CrossRefzbMATHMathSciNetGoogle Scholar - 16.B. Khosravi, B. Khosravi, and B. Khosravi, “A characterization of the finite simple group
*L*16(2) by its prime graph,”*Manuscr. Math.*, 126, 49–58 (2008).CrossRefzbMATHMathSciNetGoogle Scholar - 17.B. Khosravi, “Quasirecognition by prime graph of
*L*10(2),”*Sib. Math. J.*, 50, No. 2, 355–359 (2009).CrossRefMathSciNetGoogle Scholar - 18.B. Khosravi, “Some characterizations of
*L*9(2) related to its prime graph,”*Publ. Math. Debrecen*, 75, No. 3-4, 375–385 (2009).zbMATHMathSciNetGoogle Scholar - 19.B. Khosravi, “
*n*-Recognition by prime graph of the simple group*PSL*(2*, q*),”*J. Algebra Appl.*, 7, No. 6, 735–748 (2008).CrossRefzbMATHMathSciNetGoogle Scholar - 20.B. Khosravi and H. Moradi, “Quasirecognition by prime graph of finite simple groups
*Ln*(2) and*Un*(2),”*Acta. Math. Hung.*, 132, No. 12, 140–153 (2011).CrossRefzbMATHMathSciNetGoogle Scholar - 21.M. S. Lucido, “Prime graph components of finite almost simple groups,”
*Rend. Semin. Mat. Univ. Padova*, 102, 1–14 (1999).zbMATHMathSciNetGoogle Scholar - 22.V. D. Mazurov, “Characterizations of finite groups by the set of orders of their elements,”
*Alg. Logik.*, 36, No. 1, 23–32 (1997).CrossRefMathSciNetGoogle Scholar - 23.V. D. Mazurov and G. Y. Chen, “Recognizability of finite simple groups
*L*4(2*m*) and*U*4(2*m*) by the spectrum,”*Alg. Logik.*, 47, No. 1, 83–93 (2008).zbMATHMathSciNetGoogle Scholar - 24.W. Sierpi´nski,
*Elementary Theory of Numbers*, PWN, Warsaw (1964), Vol. 42.Google Scholar - 25.E. Stensholt, “Certain embeddings among finite groups of Lie type,”
*J. Algebra*, 53, 136–187 (1978).CrossRefzbMATHMathSciNetGoogle Scholar - 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).CrossRefMathSciNetGoogle Scholar - 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.
- 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).CrossRefMathSciNetGoogle Scholar - 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).zbMATHMathSciNetGoogle Scholar - 30.A.V. Vasil’ev and M. A. Grechkoseeva, “On the recognition of finite simple orthogonal groups of dimension 2
*m,*2*m*+1 and 2*m*+2 over the field of characteristic 2,”*Sib. Math. Zh.*, 45, No. 3, 420–431 (2004).CrossRefMathSciNetGoogle Scholar - 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).CrossRefzbMATHMathSciNetGoogle Scholar - 32.A.V. Zavarnitsin, “On the recognition of finite groups by the prime graph,”
*Alg. Logik.*, 43, No. 4, 220–231 (2006).CrossRefMathSciNetGoogle Scholar - 33.K. Zsigmondy, “Zur theorie der potenzreste,”
*Monatsh. Math. Phys.*, 3, 265–284 (1892).CrossRefzbMATHMathSciNetGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 2014