The Structure of Commutative Automorphic Loops

Details The Structure of Commutative Automorphic Loops The Structure of Commutative Automorphic Loops

2008-10-06

arxiv.org/abs/0810.1065v3

Trans. Amer. Math. Soc. 363 (2011), no. 1, 365-384

Mathematics

Group Theory

21 pages; v2: minor corrections suggested by referee; to appear in Trans. Amer. Math. Soc; v3. Sylow and Hall properties are o

Scientific paper

An \emph{automorphic loop} (or \emph{A-loop}) is a loop whose inner mappings are automorphisms. Every element of a commutative A-loop generates a group, and $(xy)^{-1} = x^{-1}y^{-1}$ holds. Let $Q$ be a finite commutative A-loop and $p$ a prime. The loop $Q$ has order a power of $p$ if and only if every element of $Q$ has order a power of $p$. The loop $Q$ decomposes as a direct product of a loop of odd order and a loop of order a power of 2. If $Q$ is of odd order, it is solvable. If $A$ is a subloop of $Q$ then $|A|$ divides $|Q|$. If $p$ divides $|Q|$ then $Q$ contains an element of order $p$. If there is a finite simple nonassociative commutative A-loop, it is of exponent 2.

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