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, $E=\mathrm{End}_R(V)$ the ring of $R$-endomorphisms of $V$, and $W$ a subgroup of $(E,+)$ such that $ab=ba$ for every $a$, $b\in W$ and $1+a$ is invertible for every $a\in W$. Then $Q_{R,V}(W)$ defined on $W\times V$ 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 $k1\cap W = 0$, naturally embedded into $\mathrm{End}_k(K)$ by $a\mapsto M_a$, $bM_a = ba$. In this case we denote the automorphic loop $Q_{R,V}(W)$ by $Q_{k<K}(W)$. We call the parameters tame if $k$ is a prime field, $W$ generates $K$ as a field over $k$, and $K$ is perfect when $\mathrm{char}(k)=2$. We describe the automorphism groups of tame automorphic loops $Q_{k<K}(W)$, and we solve the isomorphism problem for tame automorphic loops $Q_{k<K}(W)$. A special case solves a problem about automorphic loops of order $p^3$ posed by Jedli\v{c}ka, Kinyon and Vojt\v{e}chovsk\'y. We conclude the paper with a construction of an infinite $2$-generated abelian-by-cyclic automorphic loop of prime exponent.