Automorphisms of p-groups of semifield type
A projective plane may be coordinatized by a semifield if and only if it is a translation plane and the dual of a translation plane. Denote by A the group of all elations with axis the distinguished line \(\ell_{\infty}\) and by B the group of all elations with centre the distinguished point on \(\ell_{\infty}\) and put \(P=AB\). Then P satisfies (i) \(| P| =p^{3n}\), where p is a prime, and A and B are elementary abelian normal subgroups of P of order \(p^{2n}\). (ii) \(ab=ba\) implies \(a\in A\cap B\) or \(b\in A\cap B\) for all \(a\in A\), \(b\in B\). A group P with the above properties is called a p-group of semifield type. \textit{A. Cronheim} [Illinois J. Math. 9, 1-30 (1965; Zbl 0137.146)] has sown that the semifield may be recovered from P and the ordered pair (A,B). The main result of te present paper is: Let P be a p-group of semifield type of order \(p^{3n}\), p odd. Let W be the set of all abelian subgroups of P of order \(p^{2n}\). If \(| W|>2\) then: (i) \(| W| =1+p^ r\), r a divisor of n. (ii) \(PSL(2,p^ r)\leq G_{| W}\leq L_{| W}\leq P\Gamma L(2,p^ r)\) in the natural representation, where \(L=Aut(P)\) and \(G=C_ L(Z(P))\). As a corollary the author deduces that P (as an abstract group) uniquely determines up to duality the semifield plane.
scientific article; zbMATH DE number 3866071