2012/10/05 by Michael Kinyon, Kinyon, Michael, Ken Kunen +6
Engineering · Mathematics · #20N05 #FOS: Mathematics #Group Theory (math.GR) #History and Theory of Mathematics #Mathematics and Applications #graph theory and CDMA systems #math.GR #msc:20N05
paper · pdf · doi:10.48550/arxiv.1210.1642
27 pages
arxiv created 2012/10/05 · openalex publication_date 2012/10/05 · arxiv updated 2012/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops. We study uniquely 2-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman's work on uniquely 2-divisible Moufang loops) and the associated Lie rings (motivated by a construction of Wright). We prove that every automorphic loop Q of odd order is solvable, contains an element of order p for every prime p dividing |Q|, and |S| divides |Q| for every subloop S of Q. There are no finite simple nonassociative commutative automorphic loops, and there are no finite simple nonassociative automorphic loops of order less than 2500. We show that if Q is a finite simple nonassociative automorphic loop then the socle of the multiplication group of Q is not regular. The existence of a finite simple nonassociative automorphic loop remains open. Let p be an odd prime. Automorphic loops of order p or p2 are groups, but there exist nonassociative automorphic loops of order p3, some with trivial nucleus (center) and of exponent p. We construct nonassociative "dihedral" automorphic loops of order 2n for every n>2, and show that there are precisely p-2 nonassociative automorphic loops of order 2p, all of them dihedral.