On the recognizability of some projective general linear groups by the prime graph

Masoumeh Sajjadi

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 4, page 443-458
  • ISSN: 0010-2628

Abstract

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Let G be a finite group. The prime graph of G is a simple graph Γ ( G ) whose vertex set is π ( G ) and two distinct vertices p and q are joined by an edge if and only if G has an element of order p q . A group G is called k -recognizable by prime graph if there exist exactly k nonisomorphic groups H satisfying the condition Γ ( G ) = Γ ( H ) . A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that PGL ( 2 , p α ) is recognizable, if p is an odd prime and α > 1 is odd. But for even α , only the recognizability of the groups PGL ( 2 , 5 2 ) , PGL ( 2 , 3 2 ) and PGL ( 2 , 3 4 ) was investigated. In this paper, we put α = 2 and we classify the finite groups G that have the same prime graph as Γ ( PGL ( 2 , p 2 ) ) for p = 7 , 11 , 13 and 17. As a result, we show that PGL ( 2 , 7 2 ) is unrecognizable; and PGL ( 2 , 13 2 ) and PGL ( 2 , 17 2 ) are recognizable by prime graph.

How to cite

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Sajjadi, Masoumeh. "On the recognizability of some projective general linear groups by the prime graph." Commentationes Mathematicae Universitatis Carolinae 62 63.4 (2022): 443-458. <http://eudml.org/doc/299033>.

@article{Sajjadi2022,
abstract = {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.},
author = {Sajjadi, Masoumeh},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {projective general linear group; prime graph; recognition},
language = {eng},
number = {4},
pages = {443-458},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the recognizability of some projective general linear groups by the prime graph},
url = {http://eudml.org/doc/299033},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Sajjadi, Masoumeh
TI - On the recognizability of some projective general linear groups by the prime graph
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 4
SP - 443
EP - 458
AB - 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.
LA - eng
KW - projective general linear group; prime graph; recognition
UR - http://eudml.org/doc/299033
ER -

References

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  1. Akhlaghi Z., Khosravi B., Khatami M., 10.1142/S021819671000587X, Internat. J. Algebra Comput. 20 (2010), no. 7, 847–873. MR2738548DOI10.1142/S021819671000587X
  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. MR1902821
  3. Aschbacher M., Seitz G. M., 10.1017/S0027763000017438, Nagoya Math. J. 63 (1976), 1–91. MR0422401DOI10.1017/S0027763000017438
  4. Buturlakin A. A., 10.1007/s10469-008-9003-3, Algebra Logika 47 (2008), no. 2, 157–173, 264 (Russian); translation in Algebra Logic 47 (2008), no. 2, 91–99. MR2438007DOI10.1007/s10469-008-9003-3
  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. Zbl0568.20001MR0827219
  6. Darafsheh M. R., Farjami Y., Sadrudini A., A characterization property of the simple group P S L 4 ( 5 ) by the set of its element orders, Arch. Math. (Brno) 43 (2007), no. 1, 31–37. MR2310122
  7. Higman G., 10.1112/jlms/s1-32.3.335, J. Lond. Math. Soc. 32 (1957), 335–342. MR0089205DOI10.1112/jlms/s1-32.3.335
  8. Khatami M., Khosravi B., Akhlaghi Z., 10.1216/RMJ-2011-41-5-1523, Rocky Mountain J. Math. 41 (2011), no. 5, 1523–1545. MR2838076DOI10.1216/RMJ-2011-41-5-1523
  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. MR1057341
  10. Lucido M. S., Prime graph components of finite almost simple groups, Rend. Sem. Mat. Univ. Padova 102 (1999), 1–22. MR1739529
  11. Mahmoudifar A., On finite groups with the same prime graph as the projective general linear group P G L ( 2 , 81 ) , Transactions on Algebra and Its Applications 2 (2016), 43–49. MR3746331
  12. Mahmoudifar A., 10.7151/dmgaa.1256, Discuss. Math. Gen. Algebra Appl. 36 (2016), no. 2, 223–228. MR3594963DOI10.7151/dmgaa.1256
  13. Mahmoudifar A., Recognition by prime graph of the almost simple group P G L ( 2 , 25 ) , J. Linear. Topol. Algebra 5 (2016), no. 1, 63–66. MR3569945
  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. MR1454690
  15. Passman D. S., Permutation Groups, W. A. Benjamin, New York, 1968. MR0237627
  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., 10.1080/00927879408824920, Comm. Algebra 22 (1994), no. 5, 1507–1530. MR1264726DOI10.1080/00927879408824920
  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. MR3470934
  19. Shi W. J., On simple K 4 -groups, Chinese Sci. Bull. 36 (1991), no. 7, 1281–1283 (Chinese). MR1150578
  20. Simpson W. A., Frame J. S., The character tables for S L ( 3 , q ) , S L ( 3 , q 2 ) , P S L ( 3 , q ) , P S U ( 3 , q 2 ) , Canadian. J. Math. 25 (1973), no. 3, 486–494. MR0335618
  21. Srinivasan B., The characters of the finite symplectic group S p ( 4 , q ) , Trans. Am. Math. Soc. 131 (1968), no. 2, 488–525. MR0220845
  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. MR2169394
  23. Williams J. S., 10.1016/0021-8693(81)90218-0, J. Algebra 69 (1981), no. 2, 487–513. Zbl0471.20013MR0617092DOI10.1016/0021-8693(81)90218-0
  24. Yang N., Grechkoseeva M. A., Vasil'ev A. V., 10.1515/jgth-2019-0109, J. Group Theory 23 (2020), no. 3, 447–470. MR4092939DOI10.1515/jgth-2019-0109
  25. Zavarnitsine A. V., Finite simple groups with narrow prime spectrum, Sib. Èlektron. Mat. Izv. 6 (2009), 1–12. Zbl1289.20021MR2586673
  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. MR2876551
  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. MR1766731
  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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