## Linear groups generated by reflection tori.(English)Zbl 0952.20042

Let $$V$$ be a left vector space over a (possibly commutative) skew field $$k$$. For $$g\in\text{GL}(V)$$, define $$[V,g]=\{vg-v\mid v\in V\}$$ and $$C_V(g)=\{v\in V\mid vg-v=0\}$$. The subspaces $$[V,g]$$ and $$C_V(g)$$ are called center and axis of $$g$$, respectively. If $$C_V(g)$$ is a hyperplane and if $$[V,g]$$ is not contained in $$C_V(g)$$, then $$g$$ is called a reflection. Let $$H$$ be a hyperplane and $$p$$ some complement of $$H$$ in $$V$$. The group $$T_{p,H}$$ consisting of the identity and all reflections with center $$p$$ and axis $$H$$ is called a reflection torus.
The authors determine all subgroups $$G$$ of $$\text{GL}(V)$$ generated by reflection tori.
If $$G$$ is irreducible and if $$|k|>5$$, then $$G=R(V,W^*)$$, where $$W^*$$ is a subspace of $$V^*$$ with $$\text{Ann}(W^*)=0$$; the group $$G$$ contains a unique conjugacy class of reflection tori. Several reflection tori are described for $$|k|\leq 5$$.

### MSC:

 20H20 Other matrix groups over fields 51A50 Polar geometry, symplectic spaces, orthogonal spaces 20H25 Other matrix groups over rings 20E07 Subgroup theorems; subgroup growth

### Keywords:

linear groups; reflections; reflection tori; centers; axes; hyperplanes
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