A new characterization of the sporadic simple group B

This paper investigates a novel characterization method for the sporadic simple group $B$ (Baby Monster Group). Using the combination of the order component set $OC(G)$ of a finite group and the set $\pi_{p_m}(G)$ of the orders of centralizers of elements of highest order within the group, we proved: a finite group $G$ is isomorphic to the sporadic simple group $B$ if and only if the following two conditions hold: (1) $G$ shares the same largest order component $m_1(G)$ as B; (2) The sets $\pi_{p_m}(G)$ and $\pi_{p_m}(B)$ of the orders of centralizers of their respective highest-order elements are identical. By analyzing the prime graph structure, excluding the possibility of Frobenius groups and 2-Frobenius groups, and utilizing the Classification Theorem of Finite Simple Groups, we ultimately establish the sufficiency and necessity of these characterizing conditions.