On Subsequence Weighted Products

Abstract

Let $G$ be a finite group of order $n$ and let $k$ be a natural number. Let $\{x_i : i\in I\}$ be a family of elements of $G$ such that $|I|= n+k-1$. Let $v$ be the most repeated value of the family. Let $ \{ \sigma_i : 1\leq i \leq k \} $ be a family of permutations of $G$ such that $\sigma_i(1)=1$ for all $i$. We obtain the following result.

There are pairwise distinct elements $i_1, i_2, \dots ,i_k\in I$ such that \[ \prod_{1\leq j\leq k } \sigma_j \big(v^{-1}x_ {i_j }\big) =1.\]