Calculation of critical exponents for noncommutative maximal subgroups of a finite group

Non-exchangeable extremely large subgroups have important application value in the study of the nature of groups, which can reveal the intrinsic structure and behavioral patterns of groups.In this paper, a method of calculating the critical index of non-exchangeable extremal subgroups of a finite group is proposed.By introducing the Sylow subgroup of the group and the structural characteristics of the noncommutative extremal subgroup, several theorems and lemmas are proposed to derive the lower bound of the critical exponent.The validity of the method is verified by numerical simulation analysis, and the results show that the number of non-commutative extremal subgroups is significantly affected by the solvability of the group and the informality of the Sylow subgroup in different types of group structures.In the simulation, taking the karate club network as an example, the expectation value of 0.0682 is obtained through 1200 times of random graph simulation calculations, which indicates that the proposed method has high accuracy in practical applications.In addition, for the dolphin group network and the political book classification network, the simulation results also show that the calculation method can effectively reflect the intrinsic relationship and structure of the group.The study shows that the calculation method of the critical exponent of noncommutative extremely large subgroups has potential application value in finite group theory, and can provide an important basis for the classification and structure analysis of groups.