A family of groups extending McLain's

TL;DR

This paper introduces a broader family of McLain groups, exploring their structural properties and isomorphisms, extending classical results to more general partial orders.

Contribution

It constructs and analyzes a new class of McLain groups with weaker axioms, providing structural insights and novel isomorphism results.

Findings

Established a group presentation for the new family

Described factors of the descending central series

Proved a natural isomorphism involving quotient groups

Abstract

Given a strict partial order $\Delta$ on a set $\Lambda$ and an arbitrary ring $R$ with $1\neq 0$, the corresponding McLain group $M(\Delta)$ has been studied in depth. We construct a larger family of McLain groups $G(\Delta)$, where $\Delta$ is neither asymmetric nor transitive, while satisfying two weaker axioms. Structural properties common to all members~$G(\Delta)$ 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~$\Delta$, and a recursive determination of its upper central series. In addition, we prove the natural isomorphism $G(\Delta)/G(\Gamma)\cong G(\Delta\setminus\Gamma)$, where $\Gamma$ is a normal subset $\Gamma$ of $\Delta$, and $G(\Gamma)$ and $G(\Delta\setminus\Gamma)$ are extended McLain groups on their own right. This result has no parallel in the classical context.