scholarly journals Steiner triple systems with doubly transitive automorphism groups: A corollary to the classification theorem for finite simple groups

1984 ◽  
Vol 36 (1) ◽  
pp. 105-110 ◽  
Author(s):  
J.D Key ◽  
E.E Shult
Keyword(s):  
Automorphism Groups ◽  
Classification Theorem ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Transitive Automorphism

Steiner triple systems of order 19 constructed from the Steiner triple system of order 9

1983 ◽  
Vol 94 (1-2) ◽  
pp. 89-92 ◽  
Author(s):  
Alan R. Prince
Keyword(s):  
Standard Method ◽  
Triple System ◽  
Isomorphism Class ◽  
Automorphism Groups ◽  
Steiner Triple System ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Isomorphism Classes

SynopsisA standard method of constructing Steiner triple systems of order 19 from the Steiner triple system of order 9 gives rise to 212 different such systems. It is shown that there are just three isomorphism classes amongst these systems. Representatives of each isomorphism class are described and the orders of their automorphism groups are determined.

scholarly journals Free Steiner triple systems and their automorphism groups

2014 ◽  
Vol 14 (02) ◽  
pp. 1550025 ◽  
Author(s):  
Alexander Grishkov ◽  
Diana Rasskazova ◽  
Marina Rasskazova ◽  
Izabella Stuhl
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Steiner Triple Systems ◽  
Finitely Generated ◽  
Combinatorial Structures ◽  
Triple Systems ◽  
Free Objects

The paper is devoted to the study of free objects in the variety of Steiner loops and of the combinatorial structures behind them, focusing on their automorphism groups. We prove that all automorphisms are tame and the automorphism group is not finitely generated if the loop is more than 3-generated. For the free Steiner loop with three generators we describe the generator elements of the automorphism group and some relations between them.

The uniqueness of envelopes in ℵ0-categorical, ℵ0-stable structures

1984 ◽  
Vol 49 (4) ◽  
pp. 1171-1184 ◽  
Author(s):  
James Loveys
Keyword(s):  
Group Theory ◽  
Finite Subset ◽  
Classification Theorem ◽  
Simple Groups ◽  
Finite Model ◽  
Finite Simple Groups ◽  
Morley Rank ◽  
Categorical Theory ◽  
Rank 2 ◽  
Stable Structures

The Classification Theorem for ℵ0-categorical strictly minimal sets says that if H is strictly minimal and ℵ0-categorical, either H has in effect no structure at all or is essentially an affine or projective space over a finite field. Zil′ber, in [Z2], showed that if H were a counterexample to this Classification Theorem it would interpret a rank 2, degree 1 pseudoplane. Cherlin later noticed (see [CHL, Appendices 2 and 3], for the proof) that the Classification Theorem is a consequence of the Classification Theorem for finite simple groups. In [Z4] and [Z5], Zil′ber found a quite different proof of the Classification Theorem using no deep group theory.Meanwhile in [Z3], Zil′ber introduced the notion of envelope in an attempt to prove that no complete totally categorical theory T can be finitely axiomatizable. The idea of the proof was to show that if M is a model of such a T and H ⊆ M is strongly minimal, then an envelope of any sufficiently large finite subset of H is a finite model of any fixed finite subset of T. [Z3] contains an error, which Zil′ber has since corrected (in a nontrivial way).In [CHL], Cherlin, Harrington and Lachlan used the Classification Theorem to expand and reorganize Zil′ber's work. In particular, they generalized most of his work to ℵ0-categorical, ℵ0-stable structures, proved the Morley rank is finite in these structures, and introduced the powerful Coordinatization Theorem (Theorem 3.1 of [CHL]; Proposition 1.14 of the present paper). They also showed that ℵ0-categorical, ℵ0-stable structures are not finitely axiomatizable using a notion of envelope that is the same as Zil′ber's except in one particularly perverse case; [CHL]'s notion of envelope is used throughout the current paper. Peretyat'kin [P] has found an example of an ℵ1-categorical finitely axiomatizable structure.

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