On the exponent of the Weak commutativity group $\chi(G)$

TL;DR

This paper investigates the exponent of the weak commutativity group hi(G), providing bounds based on the structure of G, especially for finite solvable groups and p-groups of specific classes, revealing new divisibility properties.

Contribution

It establishes new bounds on the exponent of hi(G) for various classes of finite groups, extending understanding of their algebraic structure.

Findings

hi(G) exponent divides xp(G)^d for finite solvable groups of derived length d.

For p-groups of class p-1, hi(G) exponent divides xp(G).

Bounds depend on the prime p and the class c of the p-group, involving logarithmic expressions.

Abstract

The weak commutativity group $\chi(G)$ is generated by two isomorphic groups $G$ and $G^{\varphi }$ subject to the relations $[g,g^{\varphi}]=1$ for all $g \in G$. The group $\chi(G)$ is an extension of $D(G) = [G,G^{\varphi}]$ by $G \times G$. We prove that if $G$ is a finite solvable group of derived length $d$, then $\exp(D(G))$ divides $\exp(G)^{d}$ if $|G|$ is odd and $\exp(D(G))$ divides $2^{d-1}\cdot \exp(G)^{d}$ if $|G|$ is even. Further, if $p$ is a prime and $G$ is a $p$-group of class $p-1$, then $\exp(D(G))$ divides $\exp(G)$. Moreover, if $G$ is a finite $p$-group of class $c\geq 2$, then $\exp(D(G))$ divides $\exp(G)^{\lceil \log_{p-1}(c+1)\rceil}$ ($p\geq 3$) and $\exp(D(G))$ divides $2^{\lfloor \log_2(c)\rfloor} \cdot \exp(G)^{\lfloor \log_2(c)\rfloor+1}$ ($p=2$).