A family of groups extending McLain's

Abstract

Given a strict partial order $Δ$ on a set $Λ$ and an arbitrary ring $R$ with $1\neq 0$, the corresponding McLain group $M(Δ)$ has been studied in depth. We construct a larger family of McLain groups $G(Δ)$, where $Δ$ is neither asymmetric nor transitive, while satisfying two weaker axioms. Structural properties common to all members~$G(Δ)$ of this new family are investigated, including a group presentation, a description of the factors of its descending central series, a canonical form for its elements relative to any total order on~$Δ$, and a recursive determination of its upper central series. In addition, we prove the natural isomorphism $G(Δ)/G(Γ)\cong G(Δ\setminusΓ)$, where $Γ$ is a normal subset $Γ$ of $Δ$, and $G(Γ)$ and $G(Δ\setminusΓ)$ are extended McLain groups on their own right. This result has no parallel in the classical context.