# On group equations

Document Type : Research Paper

Authors

1 Postdoctoral Fellow, Indian Statistical Institute Bangalore, India

2 Assistant Professor in Indian Institute of Technology Delhi, India.

Abstract

Suppose $f$ is a map from a non-empty finite set $X$ to a finite group $G$. Define the map $\zeta^f_G: G\longrightarrow \mathbb{N}\cup \{0\}$ by $g\mapsto |f^{-1}(g)|$. In this article, we show that for a suitable choice of $f$, the map $\zeta^f_G$ is a character. We use our results to show that the solution function for the word equation $w(t_1,t_2,\dots,t_n)=g$ ($g\in G$) is a character, where $w(t_1,t_2,\dots,t_n)$ denotes the product of $t_1,t_2,\dots,t_n,t_1^{-1},t_2^{-1},\dots,t_n^{-1}$ in a randomly chosen order.

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### History

• Receive Date: 23 October 2013
• Revise Date: 29 April 2015
• Accept Date: 03 February 2014
• First Publish Date: 01 April 2015