Nonassociative Moufang Loops of Odd Order p1p2 2p4 Properties and Characterizations

Abstract

We investigate nonassociative Moufang loops of odd order p1p2 2q4, where p1, p2, q are distinct odd primes satisfying p1 < p2 < q. We show that there exists a normal Hall subloop of order q4 in L, and further prove that subject to some extra conditions the Moufang loop of this order is associative. Notably, the Loops Package gap4r5 does not contain Moufang loops of order p1p2 2q4, indicating a gap in the current knowledge of higher-order Moufang loops. Our results provide new insights into the structure and properties of nonassociative Moufang loops, building upon existing research.