On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure

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The nilpotentcy class for the Frobenius was determined based on the structure theorem. The socle of the groups were observed to be regular normal and elementary abelian such features were the conditions for the nilpotency classes, as they were the basis on which the socle of these groups constructed were nilpotent of some classes or order. The socle of the nilpotent groups whose structures is in conformity with D were classified based on the classification scheme for the finite primitive groups in relation to socle type. The socle type described in the classification scheme was in condition (1) was in line with the structure of D, as such it pave way in determining the socle with nilpotency class having same or similar structure with D. Further investigations showed that Frobenious group's were 2-transitive and the structure of D gave the conditions it being regular elementary abelian and so is nilpotent. It was observed the stabiliser of the groups in a finite primitive groups were paramount in the determination of the socle of the groups, as such much attention was given to the stabilizer of each group under consideration in a quest to determine the socle and the nilpotency class. The other conditions for the classification of finite primitive groups based of the socle type were not given much attention as it could give the needed condition for the existence of nilpotency class of the groups, as groups of such types were either almost simple, diagonal, product or twisted wreath product type. Therefore finite primitive group's under those conditions which could not give the expected nilpotency class and order were not give much attention. The degree of homogeneity was not given much priority as the article intended to discuss only the socle type and it nilpotency class or order.