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Square commutative groups

Weicai Wu, Mingxuan Yang, Yangbo Zhou, Chao Rong

Abstract

In this paper we first give a necessary and sufficient condition for a group $G$ generated by $n$ elements to be a square commutative group and prove $G$ is a square commutative group if and only if $\widehat{G}$ is an abelian group, then we give conditions for a group generated by two elements, with additional conditions, to be a square commutative group.

Square commutative groups

Abstract

In this paper we first give a necessary and sufficient condition for a group generated by elements to be a square commutative group and prove is a square commutative group if and only if is an abelian group, then we give conditions for a group generated by two elements, with additional conditions, to be a square commutative group.

Paper Structure

This paper contains 5 sections, 25 theorems.

Table of Contents

  1. Introduction
  2. Conditions for square commutative groups generated by two elements
  3. Square commutative groups generated by $n$ elements with $n\geq3$
  4. Relationship between square commutative groups and abelian groups
  5. The square commutativity of $BS(p,q)$ with some additional relations

Key Result

Proposition 2.2

(i) Assuming that $G$ is a square commutative group. For all $x,y\in G$, then $(xyx)^{m}y^{n}=y^{n}(xyx)^{m}$ holds for any $m,n\in \mathbb Z$. (ii) Assuming that $G$ is a free group generated by $S=\{a_{1},a_{2},\cdots, a_{n}\}$, then $G$ is a abelian group if and only if $a_{i}a_{j}=a_{j}a_{i}$ ho

Theorems & Definitions (30)

  • Definition 2.1
  • Proposition 2.2
  • Example 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Corollary 2.9
  • Theorem 2.10
  • ...and 20 more