A note on finite semifields and certain \(p\)-groups of class 2. (Q1420619)

With every (pre-)semifield \(S\) one can associate the group \(G\) generated by all translations and all shears with vertical axis of the corresponding translation plane. If \(S\) is finite of order \(p^ n\), then \(G\) is a group of order \(p^ {3n}\) of nilpotence class 2 which contains two subgroups \(X\) and \(Y\), each of order \(p^ n\), such that no nontrivial elements \(x \in X\) and \(y \in Y\) commute. The authors show that conversely every group with these properties can be obtained from a semifield in the way just described. A similar characterization of this class of groups has been given by \textit{Y. Hiramine} [Osaka J. Math. 20, 735--746 (1983; Zbl 0544.51005)], who also showed that the groups associated with two distinct semifields are isomorphic if and only if the semifields are either isotopic or one is isotopic to the dual of the other.