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تسجيل مستخدم جديدOn a question of Kulshammer for representations of finite groups in reductive groups
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Let $G$ be a simple algebraic group of type $G_2$ over an algebraically closed field of characteristic $2$. We give an example of a finite group $Gamma$ with Sylow $2$-subgroup $Gamma_2$ and an infinite family of pairwise non-conjugate homomorphisms $rhocolon Gammarightarrow G$ whose restrictions to $Gamma_2$ are all conjugate. This answers a question of Burkhard Kulshammer from 1995. We also give an action of $Gamma$ on a connected unipotent group $V$ such that the map of 1-cohomologies ${rm H}^1(Gamma,V)rightarrow {rm H}^1(Gamma_p,V)$ induced by restriction of 1-cocycles has an infinite fibre.
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