This study investigates the computational and theoretical frameworks governing Sylow p- subgroups within alternating groups An of order |An| = n!/2, focusing on their Sylow p-subgroups P ∈ Sylp(An) and structural properties, conjugacy relations, and fusion phenomena. We present novel algorithmic approaches for characterizing these subgroups through explicit construction of normalizers NAn(P) and analyze their wreath product decompositions P ≅ (Z/pZ)k ≀ H. The research employs computational group theory techniques to examine fusion systems ℱp(An) and their implications for understanding conjugacy classes. We establish new theoretical results concerning the number kp(An) of conjugacy classes of Sylow p-subgroups where pα || n! with . The research demonstrates computational complexity improvements from O(n3) to O(n2 log n) for normalizer computations. Our findings reveal structural patterns in automorphism groups Aut(P) and fusion mappings φ: P → An. The theoretical framework extends Sylow theory applications while providing O(pα log n) algorithms for researchers working with large alternating groups. Applications to group-based cryptographic protocols utilizing discrete logarithm problems in NAn(P)/CAn(P) are discussed. The methodology combines classical theorems with computational methods achieving about 60% efficiency improvements over existing algorithms.
Mohammed et al. (Tue,) studied this question.