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.
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Z. Akhlaghi, M. Khatami, and B. Khosravi, “Quasirecognition by prime graph of the simple group 2F4(q),” Acta Math. Hung., 122, No. 4, 387–397 (2009).
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).
A. Babai, B. Khosravi, and N. Hasani, “Quasirecognition by prime graph of 2Dp(3) where p = 2n +1 ≥ 5 is a prime,” Bull. Malays. Math. Sci. Soc., 32, No. 3, 343–350 (2009).
A. Babai and B. Khosravi, “Recognition by prime graph of 2D2m+1(3),” Sib. Math. J., 52, No. 5, 993–1003 (2011).
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).
M. A. Grechkoseeva,W. J. Shi, and A.V. Vasil’ev, “Recognition by spectrum of L16(2m),” Alg. Colloq., 14, No. 3, 462–470 (2007).
R. M. Guralnick and P. H. Tiep, “Finite simple unisingular groups of Lie type,” J. Group Theory, 6, 271–310 (2003).
M. Hagie, “The prime graph of a sporadic simple group,” Comm. Algebra, 31, No. 9, 4405–4424 (2003).
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).
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).
A. Khosravi and B. Khosravi, “Quasirecognition by prime graph of the simple group 2G2(q),” Sib. Math. J., 48, No. 3, 570–577 (2007).
B. Khosravi and A. Babai, “Quasirecognition by prime graph of F4(q) where q =2n >2,” Monatsh. Math., 162, No. 3, 289–296 (2011).
B. Khosravi, B. Khosravi, and B. Khosravi, “2-Recognizability of PSL(2,p2) by the prime graph,” Sib. Math. J., 49, No. 4, 749–757 (2008).
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).
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).
B. Khosravi, B. Khosravi, and B. Khosravi, “A characterization of the finite simple group L16(2) by its prime graph,” Manuscr. Math., 126, 49–58 (2008).
B. Khosravi, “Quasirecognition by prime graph of L10(2),” Sib. Math. J., 50, No. 2, 355–359 (2009).
B. Khosravi, “Some characterizations of L9(2) related to its prime graph,” Publ. Math. Debrecen, 75, No. 3-4, 375–385 (2009).
B. Khosravi, “n-Recognition by prime graph of the simple group PSL(2, q),” J. Algebra Appl., 7, No. 6, 735–748 (2008).
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).
M. S. Lucido, “Prime graph components of finite almost simple groups,” Rend. Semin. Mat. Univ. Padova, 102, 1–14 (1999).
V. D. Mazurov, “Characterizations of finite groups by the set of orders of their elements,” Alg. Logik., 36, No. 1, 23–32 (1997).
V. D. Mazurov and G. Y. Chen, “Recognizability of finite simple groups L4(2m) and U4(2m) by the spectrum,” Alg. Logik., 47, No. 1, 83–93 (2008).
W. Sierpi´nski, Elementary Theory of Numbers, PWN, Warsaw (1964), Vol. 42.
E. Stensholt, “Certain embeddings among finite groups of Lie type,” J. Algebra, 53, 136–187 (1978).
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).
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.
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).
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).
A.V. Vasil’ev and M. A. Grechkoseeva, “On the recognition of finite simple orthogonal groups of dimension 2m, 2m+1 and 2m+2 over the field of characteristic 2,” Sib. Math. Zh., 45, No. 3, 420–431 (2004).
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).
A.V. Zavarnitsin, “On the recognition of finite groups by the prime graph,” Alg. Logik., 43, No. 4, 220–231 (2006).
K. Zsigmondy, “Zur theorie der potenzreste,” Monatsh. Math. Phys., 3, 265–284 (1892).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 5, pp. 598–608, May, 2014.
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Babai, A., Khosravi, B. Groups with the Same Prime Graph as the Simple Group D n (5). Ukr Math J 66, 666–677 (2014). https://doi.org/10.1007/s11253-014-0963-2
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DOI: https://doi.org/10.1007/s11253-014-0963-2