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.

Related facts

Similar facts

• Symmetric group of degree six or higher is not weak subset-conjugacy-closed in general linear group over rationals

Opposite facts

• Brauer's permutation lemma
• Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group

Proof

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External links

• MathOverflow discussion