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Bireflectionality of the weak orthogonal and the weak symplectic groups. (English) Zbl 0533.20020

A group G is called bireflectional if every element in G is a product of two involutions in G. It has been shown earlier that the orthogonal group of a finite-dimensional regular vector space is bireflectional. The symplectic group of a finite-dimensional regular vector space is bireflectional if and only if the characteristic of the underlying field is 2. The present paper investigates similar questions in a more general setting, where the radical of the vector space V may be nontrivial and its dimension may be infinite. Let O(V) and Sp(V) be the orthogonal and symplectic group of V, respectively. Let rad V denote the radical of V and for each isometry \(\pi\) let \(F(\pi)\) be the subspace of vectors fixed under \(\pi\). Then \(O^*(V)=\{\pi \in O(V);\quad rad V\subset F(\pi)\}\) and \(Sp^*(V)=\{\pi \in Sp(V);\quad rad V\subset F(\pi)\}\) are called the weak orthogonal and the weak symplectic group, respectively. We first show that if V is finite-dimensional, then \(O^*(V)\) is bireflectional. The group \(Sp^*(V)\) is bireflectional if the characteristic of the coordinate field is 2. Let \(O^{**}(V)\) and \(Sp^{**}(V)\) be the subgroups of O(V) and Sp(V), respectively, whose elements are products of reflections and symplectic transvections, respectively. Then we show that even for infinite dimension of V the groups \(O^{**}(V)\) and \(Sp^{**}(V)\) are bireflectional.

MSC:

20H15 Other geometric groups, including crystallographic groups
51F25 Orthogonal and unitary groups in metric geometry
20F05 Generators, relations, and presentations of groups
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