Exterior Algebra and an Extension of the Feng-Sun-Xiang Theorem in $p$-groups

Let $G$ be a finite group with $|G|=p^m$ where $p$ is a prime and $m$ is a positive integer.

Let $k<p$.

Let $a_1,\ldots,a_k\in G$ be pairwise distinct and let $b_1,\ldots,b_k\in G$.

Then there exists a permutation $\sigma$ on $1,\ldots,k$ such that $a_1b_{\sigma(1)},\ldots,a_kb_{\sigma(k)}$ are pairwise distinct.

This extends a theorem of Feng, Sun and Xiang, who proved that the conclusion holds in abelian $p$-groups.