A NOVEL ALGORITHMIC FRAMEWORK FOR CLASSIFICATION OF FINITE SIMPLE GROUPS

The classification of finite simple groups (CFSG) is a cornerstone of group theory, categorizing these fundamental algebraic structures into four families: cyclic groups of prime order, alternating groups, Lie-type groups, and sporadic groups. This paper presents a novel algorithmic framework for identifying and classifying finite simple groups within a specified order range (up to 10,000). The algorithm integrates Sylow theorems, a specific divisibility condition for efficiency, and computational group theory techniques to systematically determine simplicity and classify groups into their respective categories. The paper offers a computationally optimized approach for group classification, with potential applications in cryptography, coding theory, and computational algebra.