Linear groups and collineation groups of translation planes
Since many years the author studies collineation groups of translation planes. This is equivalent with studying groups \(G\) acting on a \(2d\)-dimensional vector space over \(\text{GF}(q)\) having a partition in \(d\)-dimensional vector spaces which is invariant under \(G\). This is an important question in geometry but remarkably there are almost no results of this kind in representation theory of groups. From the geometric point of view so-called perspectives and Baer-elements play an important role. In fact an involution in \(G\) is always one of those. For these elements always the fixed space in \(V\) is half the dimension of \(V\). The author proposes to study groups acting on a vector space such that the dimension of the fixed space of nontrivial elements is a constant. In this paper he proves a weaker result. Assume \(|G|\) is even and for each involution \(i\) and \(1\neq x\in C_G(i)\) we have \(\dim C_V(i)= \dim C_V(x)\). Then he determines the possible groups \(G\). In the simple case these are \(L_2(p^a)\). If \(p\) is odd also the modules are determined, basically trivial, and 3-dimensional modules are involved in \(V\). By very clever arguments (group theory, geometry) the author proves that for \(m_2(G)\geq 3\) the centralizer of any involution is abelian and if \(m_2(G)= 2\) it is dihedral. Then it is not so hard to pin down the structure of \(G\). The remaining half of the paper then is devoted to the determination of the modules.
collineation groups of translation planes
scientific article; zbMATH DE number 1868068
scientific article