The structure of extra loops

Details The structure of extra loops The structure of extra loops

2004-04-14

arxiv.org/abs/math/0404266v3

Quasigroups and Related Systems 12 (2004), 39-60

Mathematics

Group Theory

18 pages, AMS-LaTeX; v.3: minor corrections suggested by referee

Scientific paper

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 $p$.

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