Normal Subgroups of Doubly Transitive Automorphism Groups of Chains

Abstract

We characterize the structure of the normal subgroup lattice of $2$-transitive automorphism groups $A(\Omega )$ of infinite chains $(\Omega , \leqslant )$ by the structure of the Dedekind completion $(\bar \Omega , \leqslant )$ of the chain $(\Omega , \leqslant )$. As a consequence we obtain various group-theoretical results on the normal subgroups of $A(\Omega )$, including that any proper subnormal subgroup of $A(\Omega )$ is indeed normal and contained in a maximal proper normal subgroup of $A(\Omega )$, and that $A(\Omega )$ has precisely $5$ normal subgroups if and only if the coterminality of the chain $(\Omega , \leqslant )$ is countable.