On Dowling geometries of infinite groups
نویسندگان
چکیده
A finite subgeometry of a Dowling geometry for an infinite group is exhibited, which cannot be embedded in a Dowling geometry for any finite group; this provides a negative answer to a question of Bonin. The question addressed herein is motivated by the following fundamental result of Rado, concerning the embeddability of finite geometries in projective geometries [Ra, Theorem 4]: Theorem. Every finite geometry representable over a field is representable over a finite field. In this short note we consider a corresponding embeddability question for a class of geometries known as Dowling geometries. We introduce only the essential concepts and definitions, and refer the reader to [BBB, Do, KK] for a more complete description of Dowling geometries and their properties. Let n ≥ 3 be an integer, and let A be a group. The points of the Dowling geometry Qn(A) are of two types: the joints p1, p2, . . . , pn, which form a basis; and the internal points αij (1 ≤ i < j ≤ n), where α denotes an element of A (we shall say that the point αij is “labelled” by the group element α). There are also two types of nontrivial lines: the coordinate lines, lij = {pi, pj} ∪ {αij | α ∈ A} (1 ≤ i < j ≤ n); and the transversal lines, {αij , βjk, (αβ)ik} (1 ≤ i < j < k ≤ n), each containing three points, where αβ denotes the usual product in A. (The trivial lines are those containing only two points.) In [Bo], Bonin posed the following question concerning these geometries: Question. If a finite geometry M embeds in a Dowling geometry Qn(A) for some n ≥ 3 and infinite group A, can one always find a finite group B such that M embeds in Qn(B)? Bonin points to a number of striking similarities between Dowling and projective geometries, and suggests that we may view the former as group-theoretic analogues of the latter. In view of Rado’s result, an affirmative answer to Bonin’s question would further strengthen this analogy. We shall demonstrate here that, although this is not the case, it is a purely group theoretic phenomenon that ultimately settles the matter.
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عنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 108 شماره
صفحات -
تاریخ انتشار 2004