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

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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.

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Proof

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