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A family of groups extending McLain's

Leandro Cagliero, Fernando Szechtman

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.

A family of groups extending McLain's

Abstract

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

Paper Structure

This paper contains 6 sections, 18 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Definition of the extended McLain groups
  3. An isomorphism between $G(\Delta)/G(\Gamma)$ and $G(\Delta\setminus \Gamma)$
  4. A group presentation for $G(\Delta)$
  5. A canonical form for the elements of $G(\Delta)$
  6. Upper central series of $G(\Delta)$

Key Result

Lemma 2.1

Suppose $n\geq 3$ and $i_1,\dots,i_n\in\Lambda$ satisfy: $\bullet$$(i_1,i_k)\in\Delta$ for all $1<k\leq n$, $\bullet$$(i_k,i_{k+1})\in\Delta$ for all $1\leq k<n$. Then $(i_k,i_{j})\in\Delta$ for all $1\leq k<j\leq n$. In particular, $i_1,\dots,i_n$ are all distinct. $\blacktriangleleft$$\blacktriang

Theorems & Definitions (35)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 25 more