Der Satz von Tits für PGL2(R), R ein kommutativer Ring vom stabilen Rang 2

Herzer, Armin

doi:10.1007/bf00147809

Cited by 2 publications

(4 citation statements)

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“…The converse of Herzer's theorem is also true if the ring satisfies a certain richness condition guaranteeing that (3) and (4) are fulfilled (see [7]). …”

Section: 1])mentioning

confidence: 92%

“…Our proceeding is inspired by Herzer's paper [5], where he coordinatizes certain chain spaces. Herzer himself uses the same method again in [7] for his generalization of Tits's theorem.…”

Section: Coordinatization Of the Point Setmentioning

confidence: 93%

“…Condition (5) is a version of the 3-reflection theorem. In [7] Herzer states two other technical conditions each of which may serve as a substitute for (5).…”

Section: 1])mentioning

confidence: 98%

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Projective Groups over Rings

Blunck

2002

Journal of Algebra

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In this paper, Mäurer's theorems characterizing certain subgroups of the projective group PGL 2 K over a field K are generalized to the case of rings. Key Words: projective lines over rings; projective groups over rings; permutation groups.In [9] and [10] Mäurer modifies the famous theorem due to Tits that characterizes the permutation groups PGL 2 K over commutative fields among all sharply triply transitive permutation groups. Mäurer's results are based upon the concept of symmetry instead of transitivity. The aim of the present paper is to carry over Mäurer's theorems to the case of a large class of rings, including the commutative algebras R over a field K such that n = dim K R < ∞ and K > 5n.For the reader's convenience, in Section 1 we recall the theorems by Tits and Mäurer, respectively. Moreover, we introduce the projective line over a ring and groups acting on it. We state a generalization of Tits's theorem for projective groups over rings due to Herzer.In Section 2 we find appropriate generalizations of the conditions used by Mäurer for his characterization of certain subgroups of PGL 2 K . The coordinatization results we are aiming at will require them as axioms. We are concerned with a permutation group acting on a point set and leaving invariant a certain relation on called "distant."1 The results of this paper are part of the author's Habilitationsschrift [3]. I thank the referee for his/her valuable suggestions on how to shorten Section 6. 266

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“…The converse of Herzer's theorem is also true if the ring satisfies a certain richness condition guaranteeing that (3) and (4) are fulfilled (see [7]). …”

Section: 1])mentioning

confidence: 92%

“…Our proceeding is inspired by Herzer's paper [5], where he coordinatizes certain chain spaces. Herzer himself uses the same method again in [7] for his generalization of Tits's theorem.…”

Section: Coordinatization Of the Point Setmentioning

confidence: 93%

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Projective Groups over Rings

Blunck

2002

Journal of Algebra

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“…It contains a wealth of bibliographical data. Characterisations of projective groups PGL 2 (R), where R is a ring, are given in [23], [28] and [69]. See also [40,Chapter 6].…”

Section: 53mentioning

confidence: 99%

Divisible designs, Laguerre geometry, and beyond

Havlicek

2012

J Math Sci

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Der Satz von Tits für PGL2(R), R ein kommutativer Ring vom stabilen Rang 2

Cited by 2 publications

References 5 publications

Projective Groups over Rings

Projective Groups over Rings

Divisible designs, Laguerre geometry, and beyond

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