On free involutions in linear groups and collineation groups of translation planes (Q1074860)

Let G be an even order group of linear transformations of a finite vector space V over a field of characteristic two and assume that any involution of G is free, i.e. its centralizer in V has dimension \({1/2} \dim V.\) Then G can be isomorphic to any finite group of even order, but it is difficult to determine the structure of the corresponding G-module. The authors investigate mainly the case when the ''root groups'' are big and dim V is divisible only by small powers of 2. The main result of the paper is the classification of G in the following situation. Consider a translation plane \({\mathcal T}\) of finite even order having dimension 4m over its kernel, where m is odd and let G be a group of collineations of \({\mathcal T}\) fixing an affine point. Generalizing a result of N. L. Johnson and T. G. Ostrom [\textit{N. L. Johnson}, Algebra 54, 291-315 (1978; Zbl 0401.51002)] the authors classify G in each of the cases: i) G does not contain any involutive elation, ii) G contains nontrivial elations.