Automorphic loops arising from module endomorphisms

الأصل · EN

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 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 p³ posed by Jedlička, Kinyon and Vojtěchovský. We conclude the paper with a construction of an infinite 2-generated abelian-by-cyclic automorphic loop of prime exponent.