A characterization theorem for alternating groups

In this paper, we weakened the relevant conditions of order components, and only utilized the even-order components and the orders of centralizers to investigate their impact on the group structure. We demonstrated that alternating groups with disconnected prime graphs can be uniquely determined by the even-order component $ m_1(G) $ and the set $ \pi_{p_m}(G) $. Here, $ G $ represented a finite group, $ \pi(G) $ was the set of prime factors of the order of $ G $, $ p_m $ was the largest element in $ \pi(G) $, and $ \pi_{p_m}(G) = \{ \left.|C_G(x)| \right| x\in G $ and $ |x| = p_m \}$ denoted the set of orders of centralizers of $ p_m $-order elements in $ G $.