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September 5, 2025

A new characterization of the sporadic simple group B

Key Points

A finite group is isomorphic to the sporadic simple group B under two specific conditions—the largest order component must match and centralizer orders must be identical.
The proof utilizes an analysis of the prime graph structure, while excluding options like Frobenius and 2-Frobenius groups to narrow down possibilities effectively.
Applying the Classification Theorem of Finite Simple Groups allows crucial insights into the characteristics needed for isomorphism with group B.
This new characterization opens avenues for deeper understanding of the sporadic simple group, emphasizing the significance of group properties in complex structures.

Abstract

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 ₏䂷 (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 m₁ (G) as B; (2) The sets ₏䂷 (G) and ₏䂷 (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.

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