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Dice periodic groups

Victor Petrogradsky

TL;DR

The paper introduces dice groups as a new construction of infinite finitely generated periodic groups acting on spherically homogeneous trees. It generalizes the tetrahedron group (a Gupta–Sidki–type, self-similar 2-group) to a broader family where each level is built from elementary abelian p-groups and a dice-roll rule governs recursive directed automorphisms. A central contribution is a weak but effective condition (the lucky-dice roll) under which the resulting dice group is periodic, with element orders dividing the product of a finite prime set $P$. This work extends classical constructions of periodic groups, links to self-similar and spinal-group frameworks, and yields new infinite, finitely generated periodic groups, potentially self-similar, with connections to restricted nil algebras (Phoenix algebras).

Abstract

We construct a family of finitely generated infinite periodic groups. The basic example is a 2-group, called the tetrahedron group. We generalize the construction by suggesting a family of infinite finitely generated dice groups. We provide weak conditions under which dice groups are periodic, where orders of elements are products involving finitely many given primes.

Dice periodic groups

TL;DR

The paper introduces dice groups as a new construction of infinite finitely generated periodic groups acting on spherically homogeneous trees. It generalizes the tetrahedron group (a Gupta–Sidki–type, self-similar 2-group) to a broader family where each level is built from elementary abelian p-groups and a dice-roll rule governs recursive directed automorphisms. A central contribution is a weak but effective condition (the lucky-dice roll) under which the resulting dice group is periodic, with element orders dividing the product of a finite prime set . This work extends classical constructions of periodic groups, links to self-similar and spinal-group frameworks, and yields new infinite, finitely generated periodic groups, potentially self-similar, with connections to restricted nil algebras (Phoenix algebras).

Abstract

We construct a family of finitely generated infinite periodic groups. The basic example is a 2-group, called the tetrahedron group. We generalize the construction by suggesting a family of infinite finitely generated dice groups. We provide weak conditions under which dice groups are periodic, where orders of elements are products involving finitely many given primes.

Paper Structure

This paper contains 7 sections, 8 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic example: Tetrahedron group
  3. General construction: Dice groups and their periodicity
  4. Groups acting on spherically homogeneous trees
  5. Dice groups
  6. Lucky dice roll at step $i$
  7. Periodicity of the dice groups

Key Result

Lemma 1

$A:=\langle a,w \rangle \cong \mathbf D_8$, the dihedral group of order 8.

Figures (1)

  • Figure 1: Cube vertices are identified with the set $X$. Element $w\in\mathop{\mathrm{Aut}}\nolimits T$ is defined recursively \ref{['www']}, the nontrivial sections are placed at the vertices of the red tetrahedron and marked by respective letters.

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • ...and 5 more