A new characterization of Janko simple groups

Zhangjia Han; Jiang Hu; Dongyang He; Zhangjia Han; Jiang Hu; Dongyang He

doi:10.3934/math.2024468

AIMS Mathematics

2024, Volume 9, Issue 4: 9587-9596. doi: 10.3934/math.2024468

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A new characterization of Janko simple groups

College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, China

Received: 01 December 2023 Revised: 29 February 2024 Accepted: 04 March 2024 Published: 08 March 2024
MSC : 20D10, 20D20

In this paper, we studied the influence of centralizers on the structure of groups, and demonstrated that Janko simple groups can be uniquely determined by two crucial quantitative properties: its even-order components of the group and the set $ \pi_{p_m}(G) $. Here, $ G $ represents a finite group, $ \pi(G) $ is the set of prime factors of the order of $ G $, $ p_m $ is the largest element in $ \pi(G) $, and $ \pi_{p_m}(G) = \{|C_G(x)| \large| \; x\in G $ and $ |x| = p_m \}$ denotes the set of orders of centralizers of $ p_m $-order elements in $ G $.
- finite groups,
- simple groups,
- order components,
- centralizers,
- order
Citation: Zhangjia Han, Jiang Hu, Dongyang He. A new characterization of Janko simple groups[J]. AIMS Mathematics, 2024, 9(4): 9587-9596. doi: 10.3934/math.2024468

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Abstract

In this paper, we studied the influence of centralizers on the structure of groups, and demonstrated that Janko simple groups can be uniquely determined by two crucial quantitative properties: its even-order components of the group and the set $ \pi_{p_m}(G) $. Here, $ G $ represents a finite group, $ \pi(G) $ is the set of prime factors of the order of $ G $, $ p_m $ is the largest element in $ \pi(G) $, and $ \pi_{p_m}(G) = \{|C_G(x)| \large| \; x\in G $ and $ |x| = p_m \}$ denotes the set of orders of centralizers of $ p_m $-order elements in $ G $.

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