Automorphismengruppen 8-dimensionaler Tern�rk�rper

Salzmann, Helmut

doi:10.1007/bf01214146

Cited by 25 publications

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“…This fact will provide additional information about the structure of the group F. The main tool for this procedure is the following rather technical lemma. IfT/is an arbitrary Baer ternary subfield and ifc E 7~*, then ~ = (c), otherwise we would have a tower of ternary subfields C g (c) < ~ < T and the group would be compact by [14]. Hence we have (VI) Every element c E 7-/* of a Baer ternary subfield 7-[ generates 7-[ and The proof of the lemma is divided into five steps.…”

Section: (3) Lemma Let K Be a Maximal Compact Subgroup Of P Where Fmentioning

confidence: 95%

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On the dimensions of automorphism groups of eight-dimensional ternary fields, II

Bdi

1994

Geom Dedicata

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Abstract. Let 7" be an eight-dimensional, connected, locally compact ternary field and let F denote a connected closed Lie subgroup of its automorphism group which is taken with the compact-open topology. It is proved that if the ternary fixed field .Tr of F is connected, then P is either isomorphic to one of the compact Lie groups G2 or SUaC, or the (covering) dimension of I ~ is at most 7.Mathematics Subject Classification (1991): 51A35, 12J99, 22E99. This paper continues the study of automorphism groups F of eight-dimensional locally compact connected ternary fields T in the sense of [14] and [3]. Throughout the paper we shall assume that F is a closed subgroup of the (locally compact) automorphism group of T. We shall work with the following definition of a topological temary field, compare [16, 7.2].(1) DEFINITION. A ternary field 7" = (T, 0, 1, 7-) consists of a set T containing the two distinct elements 0 and I and a temary operation 7-: T 3 ~ T: (8, z, t) 7-(8, z, t) which satisfies the following axioms: (T1) 7-(0, x, t) = 7-(x, 0, t) = t andT-(8, 1, 0) = 7-(1, 8, 0) = 8 for all 8, x, t c T.(T2) For any elements 8, x, y E T there exists a unique element t = t (s, x, y) in T such that 7-(8, x, t) = y.(T3) For any elements st, 82, tl, t2 C T with Sl ~ 82 there exist unique elements x = x(81, tl, ~2, t2) and8 = s(81, tl, 82, t2)inTsuchthatT-(si, x, ti) = y and 7-(s, 81, t) = tl holds for some y, t E Tandi = 1, 2.A temary field T is called topological if T is provided with a topology being neither discrete nor indiscrete such that the map 7-and its inverse mappings t, x and s are continuous. Using the terminology and notation of [3] the main results of the articles mentioned above are collected in the following (2) THEOREM. The following conclusions hoM:(a) dim F G 14.

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Section: (3) Lemma Let K Be a Maximal Compact Subgroup Of P Where Fmentioning

confidence: 95%

“…This paper continues the study of automorphism groups F of eight-dimensional locally compact connected ternary fields T in the sense of [14] and [3]. Throughout the paper we shall assume that F is a closed subgroup of the (locally compact) automorphism group of T. We shall work with the following definition of a topological temary field, compare [16, 7.2].…”

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confidence: 99%

On the dimensions of automorphism groups of eight-dimensional ternary fields, II

Bdi

1994

Geom Dedicata

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“…[] (8.20) REMARK. Using a more elaborate compactness criterion (see [21]) H. Salzmann excluded the group ASU3C in the case of projective planes. Our attempts to generalize this criterion to the ease of stable planes did not succeed, however.…”

Section: Semi-planar Groups Of Eight-dimensional Planesmentioning

confidence: 99%

“…The main source of inspiration has been the treatment of stabilizers of quadrangles in compact connected projective planes by H. Salzmann [20: section 2], [21]. The author was introduced to the study of stable planes by H. Salzmann and R. LSwen, and also owes thanks to 1%.…”

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Planar groups of automorphisms of stable planes

Stroppel

1992

J Geom

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“…Let A be the stabilizer of a quadrangle in a 16-dimensional projective plane. Then, dimA -__6 14, [13] Satz, and dimA -> 12 implies A ~ G2, [2] (4.1). If A ~ Gz, then the subplane consisting of the fixed points of A is two-dimensional, [18] 96.35.…”

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Large semisimple groups on 16-dimensional compact projective planes are almost simple

Priwitzer

1997

Arch. Math.

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The paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E6(-26). A 16-dimensional, compact projective plane ~ admitting an automorphism group of dimension 41 or more is classical, [18] 87.5 and 87.7. For the special case of a semisimple group d acting on ~ the same result can be obtained if dimd _-> 37, see [i6]. Our aim is to lower this bound. We show: if d is semisimple and dim d _-> 29, then ~ is either classical or a Moufang-Hughes plane or A is isomorphic to Spin 9 (~., r), r E {0, 1}.The underlying paper contains the first part of the proof showing that A is in fact almost simple.O. Introduction. Let ~ be a 16-dimensional compact projective plane with point space P and line space ~. An example of such a plane is the Moufang plane ~2 9 over the octonions, the so-called classical plane of dimension 16. In order to classify such planes we consider a subgroup d of the automorphism group Aut(~) of ~'. In the compact-open topology, Aut (~) is a locally compact transformation group, [18] 44.3. The group A shall be closed and connected. The larger the topological dimension of A, the better are the chances to obtain a complete classification of planes ~ with d < Aut (~).We also take into account the structure of A. Semisimple and non-semisimple groups behave in a completely different way. For projective planes the main difference between these two types is based on the fact that semisimple groups (with one exception) always contain involutions. Here, we will restrict the discussion to semisimple groups. Now we can formulate the problem: classify all 16-dimensional compact projective planes ~' admitting a semisimple group A < Aut (~') with dimA => d for some suitably chosen d. In [16] this problem was solved for d = 37. In this case ~ is always classical. In the present paper and in [9] we will lower this bound to d = 29. Our aim is to prove the following Theorem. Let ~ be a 16-dimensional compact projective plane and .4 a connected and closed subgroup of the automorphism group of ~. If .4 is semisimple and at least 29-dimensional, then one of the following statements holds:Mathematics Subject Classification (1991): 51H10.

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Automorphismengruppen 8-dimensionaler Tern�rk�rper

Cited by 25 publications

References 20 publications

On the dimensions of automorphism groups of eight-dimensional ternary fields, II

On the dimensions of automorphism groups of eight-dimensional ternary fields, II

Planar groups of automorphisms of stable planes

Large semisimple groups on 16-dimensional compact projective planes are almost simple

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