Automorphic loops arising from module endomorphisms

Abstract

A loop is automorphic if all its inner mappings are automorphisms. We construct a large family of automorphic loops as follows. Let R be a commutative ring, V an R-module, E = EndR(V ) the ring of R-endomorphisms of V , and W a subgroup of (E,+) such that ab = ba for every a, b ∈W and 1+a is invertible for every a ∈W . Then QR,V (W ) defined on W × V by (a, u)(b, v) = (a + b, u(1 + b) + v(1− a)) is an automorphic loop. A special case occurs when R = k < K = V is a field extension and W is a k-subspace of K such that k1 ∩W = 0, naturally embedded into Endk(K) by a 7→ Ma, bMa = ba. In this case we denote the automorphic loop QR,V (W ) by Qk<K(W ). We call the parameters tame if k is a prime field, W generates K as a field over k, and K is perfect when char(k) = 2. We describe the automorphism groups of tame automorphic loops Qk<K(W ), and we solve the isomorphism problem for tame automorphic loops Qk<K(W ). A special case solves a problem about automorphic loops of order p posed by Jedlicka, Kinyon and Vojtěchovský. We conclude the paper with a construction of an infinite 2-generated abelian-by-cyclic automorphic loop of prime exponent.