A New Quantitative Characterization of the Monster Group and the Baby Monster Group

Let G be a finite group, π(G) the set of prime divisors of the order of G, Pm the greatest element in π(G), and πpm(G) the set of orders of centralizers of elements of pm maximal order in G. This paper presents a novel and comprehensive quantitative characterization of the Monster group F1 and the Baby Monster group F2, achieved through the employment of the even-order component inherent to a group, in conjunction with the aggregate set comprising the orders of centralizers associated with its elements of maximal order. We establish that a finite group G, under the specified conditions, is necessarily isomorphic to the sporadic simple group M under consideration, thereby furnishing a rigorous identification within the classification framework of finite simple groups. if and only if the following two conditions hold: (1) the even-order component of m1 is identical to that of G; (2) πpm(G)= πpm(M), where M is either the Monster group F1 or the Baby Monster group F2 .