%0 Journal Article %T On group equations %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A Prajapati, S. K. %A Sarma, R. %D 2015 %\ 04/01/2015 %V 41 %N 2 %P 315-324 %! On group equations %K finite groups %K word equations %K group characters %R %X  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.   %U http://bims.iranjournals.ir/article_611_12adc2b3a8b377f26a5c5fcccd2a6e5e.pdf