Doubly transitive groups in which involutions fix one or three points

1969 ◽  
Vol 111 (4) ◽  
pp. 311-321 ◽  
Author(s):  
Jennifer D. King
Keyword(s):  
Transitive Groups ◽  
Doubly Transitive Groups

Doubly transitive groups and lattices

1999 ◽  
Vol 93 (6) ◽  
pp. 809-823 ◽  
Author(s):  
K. S. Abdukhalikov
Keyword(s):  
Transitive Groups ◽  
Doubly Transitive Groups

scholarly journals On some doubly transitive groups

1971 ◽  
Vol 17 (3) ◽  
pp. 437-450 ◽  
Author(s):  
Koichiro Harada
Keyword(s):  
Transitive Groups ◽  
Doubly Transitive Groups

scholarly journals Pseudo-fields and doubly transitive groups

1972 ◽  
Vol 7 (2) ◽  
pp. 163-168
Author(s):  
F.W. Wilke
Keyword(s):  
Near Field ◽  
Transitive Group ◽  
Affine Transformations ◽  
Additive System ◽  
Transitive Groups ◽  
Open Question ◽  
Doubly Transitive Groups

A sharply doubly transitive group which acts on a set of at least two elements is isomorphic to the group of affine transformations on a system S. This statement is true if S is replaced by either strong pseudo-field or pseudo-field. The additive system of a strong pseudo-field is a loop while the additive system of a pseudo-field need not be a loop. We show that any pseudo-field is either a strong pseudo-field or can be obtained from a strong pseudo-field in a nice way. Every near-field is a strong pseudo-field. The converse is an open question.

On a Class of Doubly Transitive Groups: II

1964 ◽  
Vol 79 (3) ◽  
pp. 514 ◽  
Author(s):  
Michio Suzuki
Keyword(s):  
Transitive Groups ◽  
Doubly Transitive Groups

On Finite Groups of the Form ABA

1962 ◽  
Vol 14 ◽  
pp. 195-236 ◽  
Author(s):  
Daniel Gorenstein
Keyword(s):  
Finite Groups ◽  
General Result ◽  
Simple Groups ◽  
Normal Complement ◽  
Transitive Groups ◽  
Doubly Transitive Groups

The class of finite groups G of the form ABA, where A and B are subgroups of G, is of interest since it includes the finite doubly transitive groups, which admit such a representation with A the subgroup fixing a letter and B of order 2. It is natural to ask for conditions on A and B which will imply the solvability of G. It is known that a group of the form AB is solvable if A and B are nilpotent. However, no such general result can be expected for ABA -groups, since the simple groups PSL(2,2n) admit such a representation with A cyclic of order 2n + 1 and B elementary abelian of order 2n. Thus G need not be solvable even if A and B are abelian.In (3) Herstein and the author have shown that G is solvable if A and B are cyclic of relatively prime orders; and in (2) we have shown that G is solvable if A and B are cyclic and A possesses a normal complement in G.

scholarly journals Transitive Extensions of a Class of Doubly Transitive Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 159-169 ◽  
Author(s):  
Michio Suzuki
Keyword(s):  
Recent Result ◽  
Permutation Group ◽  
Unitary Group ◽  
Simple Groups ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Precise Statement ◽  
Transitive Groups ◽  
Recent Theorem ◽  
Doubly Transitive Groups

1. When a permutation group G on a set Ω is given, a transitive extension G of G is defined to be a transitive permutation group on the set Γ which is a union of Ω and a new point ∞ such that the stabilizer of ∞ in G1 is isomorphic to G as a permutation group on Ω. The purpose of this paper is to prove that many known simple groups which can be represented as doubly transitive groups admit no transitive extension. Precise statement is found in Theorem 2. For example, the simple groups discovered by Ree [5] do not admit transitive extensions. Theorem 2 includes also a recent result of D. R. Hughes [3] which states that the unitary group U3(q) q>2 does not admit a transitive extension. As an application we prove a recent theorem of H. Nagao [4], which generalizes a theorem of Wielandt on the non-existence of 8-transitive permutation groups not containing the alternating groups under Schreier’s conjecture.

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