Recognition of the symplectic simple group $ PSp_4(p) $ by the order and degree prime-power graph

Chao Qin; Yu Li; Zhongbi Wang; Guiyun Chen; Chao Qin; Yu Li; Zhongbi Wang; Guiyun Chen

doi:10.3934/math.2024139

AIMS Mathematics

2024, Volume 9, Issue 2: 2808-2823. doi: 10.3934/math.2024139

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Recognition of the symplectic simple group $ PSp_4(p) $ by the order and degree prime-power graph

1.
College of Science, Chongqing University of Posts and Telecommunications, Nanan, Chongqing 400065, China
2.
School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China
3.
School of Mathematics and Statistics, Chongqing Technology and Business University, Nanan, Chongqing 400067, China
4.
School of Mathematics and Statistics, Southwest University, Beibei, Chongqing 400715, China

Received: 02 November 2023 Revised: 22 December 2023 Accepted: 25 December 2023 Published: 29 December 2023
MSC : 20D05, 20C15, 20D35

Let $ G $ be a finite group, $ \operatorname{cd}(G) $ the set of all irreducible character degrees of $ G $, and $ \rho(G) $ the set of all prime divisors of integers in $ \operatorname{cd}(G) $. For a prime $ p $ and a positive integer $ n $, let $ n_p $ denote the $ p $-part of $ n $. The degree prime-power graph of $ G $ is a graph whose vertex set is $ V(G) = \left\{p^{e_p(G)} \mid p \in \rho(G)\right\} $, where $ p^{e_p(G)} = \max \left\{n_p \mid n \in \operatorname{cd}(G)\right\} $, and there is an edge between distinct numbers $ x, y \in V(G) $ if $ x y $ divides some integer in $ \operatorname{cd}(G) $. The authors have previously shown that some non-abelian simple groups can be uniquely determined by their orders and degree prime-power graphs. In this paper, the authors build on this work and demonstrate that the symplectic simple group $ PSp_4(p) $ can be uniquely identified by its order and degree prime-power graph.
- degree prime-power graph,
- irreducible character,
- degree,
- maximal $ p $-part,
- simple group,
- subnormal series
Citation: Chao Qin, Yu Li, Zhongbi Wang, Guiyun Chen. Recognition of the symplectic simple group $ PSp_4(p) $ by the order and degree prime-power graph[J]. AIMS Mathematics, 2024, 9(2): 2808-2823. doi: 10.3934/math.2024139

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Abstract

Let $ G $ be a finite group, $ \operatorname{cd}(G) $ the set of all irreducible character degrees of $ G $, and $ \rho(G) $ the set of all prime divisors of integers in $ \operatorname{cd}(G) $. For a prime $ p $ and a positive integer $ n $, let $ n_p $ denote the $ p $-part of $ n $. The degree prime-power graph of $ G $ is a graph whose vertex set is $ V(G) = \left\{p^{e_p(G)} \mid p \in \rho(G)\right\} $, where $ p^{e_p(G)} = \max \left\{n_p \mid n \in \operatorname{cd}(G)\right\} $, and there is an edge between distinct numbers $ x, y \in V(G) $ if $ x y $ divides some integer in $ \operatorname{cd}(G) $. The authors have previously shown that some non-abelian simple groups can be uniquely determined by their orders and degree prime-power graphs. In this paper, the authors build on this work and demonstrate that the symplectic simple group $ PSp_4(p) $ can be uniquely identified by its order and degree prime-power graph.

References

[1]	B. Huppert, Some simple groups which are determined by the set of their character degrees I, Illinois J. Math., 44 (2000), 828–842. http://dx.doi.org/10.1215/ijm/1255984694 doi: 10.1215/ijm/1255984694
[2]	H. Behravesh, M. Ghaffarzadeh, M. Ghasemi, S. Hekmatara, Recognition of Janko groups and some simple $K_4$-groups by the order and one irreducible character degree or character degree graph, Int. J. Group Theory, 10 (2021), 1–10. http://dx.doi.org/10.22108/IJGT.2019.113029.1502 doi: 10.22108/IJGT.2019.113029.1502
[3]	S. Heydari, N. Ahanjideh, Characterization of some simple $K_4$-groups by some irreducible complex character degrees, Int. J. Group Theory, 5 (2016), 61–74. http://dx.doi.org/10.22108/IJGT.2016.8233 doi: 10.22108/IJGT.2016.8233
[4]	B. Khosravi, B. Khosravi, B. Khosravi, Z. Momen, A new characterization for the simple group $PSL(2, p^2)$ by order and some character degrees, Czech. Math. J., 65 (2015), 271–280. http://dx.doi.org/10.1007/s10587-015-0173-6 doi: 10.1007/s10587-015-0173-6
[5]	H. Xu, G. Chen, Y. Yan, A new characterization of simple $K_3$-groups by their orders and large degrees of their irreducible characters, Commun. Algebra, 42 (2014), 5374–5380. http://dx.doi.org/10.1080/00927872.2013.842242 doi: 10.1080/00927872.2013.842242
[6]	H. Xu, Y. Yan, G. Chen, A new characterization of Mathieu-groups by the order and one irreducible character degree, J. Inequal. Appl., 2013 (2013), 209. http://dx.doi.org/10.1186/1029-242X-2013-209 doi: 10.1186/1029-242X-2013-209
[7]	M. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math., 38 (2008), 175–211. http://dx.doi.org/10.1216/RMJ-2008-38-1-175 doi: 10.1216/RMJ-2008-38-1-175
[8]	B. Khosravi, B. Khosravi, B. Khosravi, Z. Momen, Recognition of some simple groups by character degree graph and order, Math. Reports, 18 (2016), 51–61.
[9]	S. Heydari, N. Ahanjideh, Some simple groups which are determined by their character degree graphs, Sib. Elektron. Math. Re., 13 (2016), 1290–1299. http://dx.doi.org/10.17377/semi.2016.13.101 doi: 10.17377/semi.2016.13.101
[10]	C. Qin, Y. Yan, K. Shum, G. Chen, Mathieu groups and its degree prime-power graphs, Commun. Algebra, 47 (2019), 4173–4180. http://dx.doi.org/10.1080/00927872.2019.1579342 doi: 10.1080/00927872.2019.1579342
[11]	C. Qin, Research on the sporadic simple groups and their degree prime-power graphs of irreducible character degrees, Ph.D thesis, Southwest University, 2019.
[12]	Z. Wang, C. Qin, H. Lv, Y. Yan, G. Chen, A new characterization of $L_2(p^2)$, Open Math., 18 (2020), 907–915. http://dx.doi.org/10.1515/math-2020-0048 doi: 10.1515/math-2020-0048
[13]	Z. Wang, G. Chen, A new characterization of $L_2(p^3)$, Commun. Algebra, 50 (2022), 4000–4008. http://dx.doi.org/10.1080/00927872.2022.2057509 doi: 10.1080/00927872.2022.2057509
[14]	J. Conway, R. Curtis, S. Norton, R. Parker, R. Wilson, Atlas of finite groups, New York: Oxford University Press, 1985.
[15]	M. Shahabi, H. Mohtadifar, The characters of finite projective symplectic group $PSp(4, q)$, In: Groups St Andrews 2001 in Oxford, Cambridge: Cambridge University Press, 2003,496–527. http://dx.doi.org/10.1017/CBO9780511542787.018
[16]	I. Isaacs, Character theory of finite groups, New York: Academic Press, 1976.
[17]	H. Tong-Viet, The simple Ree groups ${}^2F_4(q^2)$ are determined by the set of their character degrees, J. Algebra, 339 (2011), 357–369, http://dx.doi.org/10.1016/j.jalgebra.2011.04.034 doi: 10.1016/j.jalgebra.2011.04.034
[18]	D. White, Character degrees of extensions of $PSL_2(q)$ and $SL_2(q)$, J. Group Theory, 16 (2013), 1–33. http://dx.doi.org/10.1515/jgt-2012-0026 doi: 10.1515/jgt-2012-0026

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