2004/04/30 by Michael K. Kinyon, Kenneth Kunen
Mathematics · #math.GR #msc:20N05
published as Quasigroups and Related Systems 12 (2004), 39-60 · 18 pages, AMS-LaTeX; v.3: minor corrections suggested by referee
arxiv created 2004/07/08 · arxiv updated 2009/12/01
The Sylow theorems hold for finite extra loops, as does P. Hall's theorem for finite solvable extra loops. Every finite nonassociative extra loop Q has a nontrivial center, Z(Q). Furthermore, Q/Z(Q) is a group whenever |Q| < 512. Loop extensions are used to construct an infinite nonassociative extra loop with a trivial center and a nonassociative extra loop Q of order 512 such that Q/Z(Q) is nonassociative. There are exactly 16 nonassociative extra loops of order 16p for each odd prime p.