# Analogue of Brauer's permutation lemma fails over rationals for every non-cyclic finite group

## Statement

Let $G$ be a finite group that is not a cyclic group (in particular, not a finite cyclic group). Then, there exists two permutation representation $\varphi_1, \varphi_2:G \to S_n$ such that $\varphi_1$ and $\varphi_2$ are equivalent as linear representations over $\mathbb{Q}$ (viewed this way by the embedding of $S_n$ in $GL(n,\mathbb{Q})$ as permutation matrices) but are not equivalent as permutation representations.

In other words, the analogue of Brauer's permutation lemma fails to hold for non-cyclic finite groups.