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A saturation theorem for submonoids of nilpotent groups and the Identity Problem

Doron Shafrir

Abstract

If $M$ is a submonoid of a finitely generated nilpotent group $G$, and $MG'$ is a finite index subgroup of $G$, then $M$ itself is a finite index subgroup of $G$. If $MG'=G$, then $M=G$. This generalizes a well-known theorem for subgroups of finitely generated nilpotent groups. As a result, we give an algorithm for the Identity Problem in nilpotent groups.

A saturation theorem for submonoids of nilpotent groups and the Identity Problem

Abstract

If is a submonoid of a finitely generated nilpotent group , and is a finite index subgroup of , then itself is a finite index subgroup of . If , then . This generalizes a well-known theorem for subgroups of finitely generated nilpotent groups. As a result, we give an algorithm for the Identity Problem in nilpotent groups.
Paper Structure (6 sections, 7 theorems, 8 equations, 1 algorithm)

This paper contains 6 sections, 7 theorems, 8 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. A saturation theorem
  3. Related results
  4. Notations
  5. A saturation theorem for submonoids of nilpotent groups
  6. The Identity Problem for nilpotent groups

Key Result

Proposition 1

if $x_1,x_2,y\in G$ and $[x_1,y]\in Z(G)$ then $[x_1x_2,y]=[x_1,y][x_2,y]$ and $[y,x_1x_2]=[y,x_1][y,x_2]$

Theorems & Definitions (14)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 2
  • proof
  • ...and 4 more