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The generalized André systems \(A(F,\Gamma,(g_ i),(f_ i),\in)\). (English) Zbl 0695.51001
Given a field \(Q=(Q,+,\cdot)\), a map \(\phi\) : \(Q^*\to Aut Q\); \(x\to \phi_ x\) \((Q^*:=Q\setminus \{0\}\), Aut Q: group of automorphisms of Q) is called an automorphic derivation map of Q, if \(\phi_ 1=id\) and for every pair (a,b) of elements of \(Q^*\) there exists one and only one \(x\in Q^*\) with \(x\phi_ x(a)=b\). Defining \(a\circ b:=a\phi_ a(b)\) if \(a\neq 0\) and \(0\circ b=0\), \(Q^{\phi}:=(Q,+,\circ)\) becomes a quasifield, often called a generalized André-system, i.e. \((Q,+)\) is an abelian group, \((Q^*,\circ)\) is a loop with identity 1, \(0\circ a=0\) and \(a\circ (b+c)=a\circ b+a\circ c\) for all a,b,c\(\in Q\). A quasifield is said to be planar, if the equation \(a\circ x=b\circ x+c\) has a solution whenever \(a\neq b\). Planarity is necessary and sufficient for quasifields to coordinatize translation planes. The authors call planar quasifields simply quasifields, while not necessarily planar quasifield are attributed weak. They give a slightly circumstantial proof of the fact that \(Q^{\phi}\) is planar and that the fixed field K of \(\phi (Q^*)\) in Q equals the kernel of Q if Q is algebraic over K.
An extensive class of automorphic derivations is defined. After some remarks how to construct explicitly derivations of this kind, the authors proceed in showing that the following known classes of quasifields are contained in their class: Lüneburg’s generalization of André- quasifields, \(Q^{\phi}\) whenever Q/K is a cyclic extension of prime power degree, the finite Dickson-nearfields and the class of finite quasifields \(Q_ g\) constructed by Foulser.
The last section deals with the collineation groups of the corresponding translation planes in certain special cases.
Reviewer: H.Kiechle

MSC:
51A25 Algebraization in linear incidence geometry
12K99 Generalizations of fields
51A40 Translation planes and spreads in linear incidence geometry
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