A new characterization of Janko simple groups

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 $.