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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.
Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $pgeq 0$. We show that if $H_1$ and $H_2$ are connected subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent subgroup and $H_1/R_u(H_1)$ an
We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, whi
Let $G$ be a reductive algebraic group---possibly non-connected---over a field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a degeneration process for obtaining from $H$ a completely reducible subgroup $H$ of $G$; one takes a limi
Let $G$ be a reductive algebraic group over an algebraically closed field and let $V$ be a quasi-projective $G$-variety. We prove that the set of points $vin V$ such that ${rm dim}(G_v)$ is minimal and $G_v$ is reductive is open. We also prove some r
Let $F$ be either $mathbb{R}$ or a finite extension of $mathbb{Q}_p$, and let $G$ be a finite central extension of the group of $F$-points of a reductive group defined over $F$. Also let $pi$ be a smooth representation of $G$ (Frechet of moderate gro