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Keywords:
projective general linear group; prime graph; recognition
projective general linear group; prime graph; recognition
Summary:
Let $G $ be a finite group. The prime graph of $G$ is a simple graph $\Gamma(G)$ whose vertex set is $\pi(G)$ and two distinct vertices $p$ and $q$ are joined by an edge if and only if $G$ has an element of order $pq$. A group $ G $ is called $ k $-recognizable by prime graph if there exist exactly $ k$ nonisomorphic groups $ H$ satisfying the condition $ \Gamma(G) = \Gamma(H)$. A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that ${\rm PGL}(2,p^\alpha) $ is recognizable, if $ p$ is an odd prime and $ \alpha > 1$ is odd. But for even $ \alpha $, only the recognizability of the groups $ {\rm PGL}(2, 5^2)$, $ {\rm PGL}(2, 3^2) $ and $ {\rm PGL}(2, 3^4) $ was investigated. In this paper, we put $ \alpha = 2$ and we classify the finite groups $G$ that have the same prime graph as $\Gamma({\rm PGL}(2, p^2))$ for $p=7, 11, 13$ and 17. As a result, we show that ${\rm PGL}(2, 7^2)$ is unrecognizable; and ${\rm PGL}(2, 13^2)$ and ${\rm PGL}(2, 17^2)$ are recognizable by prime graph.
References:
[1] Akhlaghi Z., Khosravi B., Khatami M.: Characterization by prime graph of $ PGL(2,p^{k})$ where $p$ and $k>1$ are odd. Internat. J. Algebra Comput. 20 (2010), no. 7, 847–873. DOI 10.1142/S021819671000587X | MR 2738548
[2] Aleeva M. R.: On composition factors of finite groups having the same set of element orders as the group $U_3(q)$. Sibirsk. Mat. Zh. 43 (2002), no. 2, 249–267 (Russian); translation in Siberian Math. J. 43 (2002), no. 2, 195–211. MR 1902821
[3] Aschbacher M., Seitz G. M.: Involutions in Chevalley groups over fields of even order. Nagoya Math. J. 63 (1976), 1–91. DOI 10.1017/S0027763000017438 | MR 0422401
[4] Buturlakin A. A.: Spectra of finite linear and unitary groups. Algebra Logika 47 (2008), no. 2, 157–173, 264 (Russian); translation in Algebra Logic 47 (2008), no. 2, 91–99. DOI 10.1007/s10469-008-9003-3 | MR 2438007
[5] Conway J. H., Curtis R. T., Norton S. P., Parker R. A., Wilson R. A.: Atlas of Finite Groups. Clarendon Press (Oxford), London, 1985. MR 0827219 | Zbl 0568.20001
[6] Darafsheh M. R., Farjami Y., Sadrudini A.: A characterization property of the simple group $ PSL_4(5)$ by the set of its element orders. Arch. Math. (Brno) 43 (2007), no. 1, 31–37. MR 2310122
[7] Higman G.: Finite groups in which every element has prime power order. J. Lond. Math. Soc. 32 (1957), 335–342. DOI 10.1112/jlms/s1-32.3.335 | MR 0089205
[8] Khatami M., Khosravi B., Akhlaghi Z.: NCF-distinguishability by prime graph of $PGL(2, p)$, where $p$ is a prime. Rocky Mountain J. Math. 41 (2011), no. 5, 1523–1545. DOI 10.1216/RMJ-2011-41-5-1523 | MR 2838076
[9] Kleidman P. B., Liebeck M. W.: The Subgroup Structure of the Finite Classical Groups. London Mathematical Society Lecture Note Series, 129, Cambridge University Press, Cambridge, 1990. MR 1057341
[10] Lucido M. S.: Prime graph components of finite almost simple groups. Rend. Sem. Mat. Univ. Padova 102 (1999), 1–22. MR 1739529
[11] Mahmoudifar A.: On finite groups with the same prime graph as the projective general linear group $PGL(2, 81)$. Transactions on Algebra and Its Applications 2 (2016), 43–49. MR 3746331
[12] Mahmoudifar A.: On the unrecognizability by prime graph for the almost simple group $ PGL(2, 9)$. Discuss. Math. Gen. Algebra Appl. 36 (2016), no. 2, 223–228. DOI 10.7151/dmgaa.1256 | MR 3594963
[13] Mahmoudifar A.: Recognition by prime graph of the almost simple group $ PGL(2, 25)$. J. Linear. Topol. Algebra 5 (2016), no. 1, 63–66. MR 3569945
[14] Mazurov V. D.: Characterization of finite groups by sets of orders of their elements. Algebra i Logika 36 (1997), no. 1, 37–53, 117 (Russian); translation in Algebra and Logic 36 (1997), no. 1, 23–32. MR 1454690
[15] Passman D. S.: Permutation Groups. W. A. Benjamin, New York, 1968. MR 0237627
[16] Perumal P.: On the Theory of the Frobenius Groups. Ph.D. Dissertation, University of Kwa-Zulu Natal, Pietermaritzburg, 2012.
[17] Praeger C. E., Shi W. J.: A characterization of some alternating and symmetric groups. Comm. Algebra 22 (1994), no. 5, 1507–1530. DOI 10.1080/00927879408824920 | MR 1264726
[18] Sajjadi M., Bibak M., Rezaeezadeh G. R.: Characterization of some projective special linear groups in dimension four by their orders and degree patterns. Bull. Iranian Math. Soc. 42 (2016), no. 1, 27–36. MR 3470934
[19] Shi W. J.: On simple K$4$-groups. Chinese Sci. Bull. 36 (1991), no. 7, 1281–1283 (Chinese). MR 1150578
[20] Simpson W. A., Frame J. S.: The character tables for $ SL(3, q)$, $ SL(3, q^2)$, $ PSL(3, q)$, $ PSU(3, q^2)$. Canadian. J. Math. 25 (1973), no. 3, 486–494. MR 0335618
[21] Srinivasan B.: The characters of the finite symplectic group $ Sp(4,q)$. Trans. Am. Math. Soc. 131 (1968), no. 2, 488–525. MR 0220845
[22] Vasil'ev A. V., Grechkoseeva M. A.: On recognition by spectrum of finite simple linear groups over fields of characteristic $2$. Sibirsk. Mat. Zh. 46 (2005), no. 4, 749–758 (Russian); translation in Siberian Math. J. 46 (2005), no. 4, 593–600. MR 2169394
[23] Williams J. S.: Prime graph components of finite groups. J. Algebra 69 (1981), no. 2, 487–513. DOI 10.1016/0021-8693(81)90218-0 | MR 0617092 | Zbl 0471.20013
[24] Yang N., Grechkoseeva M. A., Vasil'ev A. V.: On the nilpotency of the solvable radical of a finite group isospectral to a simple group. J. Group Theory 23 (2020), no. 3, 447–470. DOI 10.1515/jgth-2019-0109 | MR 4092939
[25] Zavarnitsine A. V.: Finite simple groups with narrow prime spectrum. Sib. Èlektron. Mat. Izv. 6 (2009), 1–12. MR 2586673 | Zbl 1289.20021
[26] Zavarnitsine A. V.: Fixed points of large prime-order elements in the equicharacteristic action of linear and unitary groups. Sib. Èlektron. Mat. Izv. 8 (2011), 333–340. MR 2876551
[27] Zavarnitsine A. V., Mazurov V. D.: Element orders in coverings of symmetric and alternating groups. Algebra Log. 38 (1999), no. 3, 296–315, 378 (Russian); translation in Algebra and Logic 38 (1999), no. 3, 159–170. MR 1766731
[28] HASH(0x126bfa8): The GAP Group, GAP – Groups, Algorithms, and Programming, Version $4.11.1$. 2021, https://www.gap-system.org
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