A geometric consequence of the classification of finite doubly transitive groups

1986 ◽  
Vol 21 (2) ◽  
Author(s):  
Anne Delandtsheer
Keyword(s):  
Transitive Groups ◽  
Doubly Transitive Groups ◽  
Geometric Consequence

scholarly journals On rank5projective planes

1984 ◽  
Vol 7 (2) ◽  
pp. 327-338
Author(s):  
Otto Bachmann
Keyword(s):  
Collineation Group ◽  
Projective Planes ◽  
Low Rank ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Finite Degree ◽  
Transitive Groups ◽  
Collineation Groups ◽  
Doubly Transitive Groups

In this paper we continue the study of projective planes which admit collineation groups of low rank (Kallaher [1] and Bachmann [2,3]). A rank5collineation group of a projective planeℙof ordern≠3is proved to be flag-transitive. As in the rank3and rank4case this implies that isℙnot desarguesian and thatnis (a prime power) of the formm4ifmis odd andn=m2withm≡0mod4ifnis even. Our proof relies on the classification of all doubly transitive groups of finite degree (which follows from the classification of all finite simple 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 Finite groups in which some property of two-generator subgroups is transitive

2007 ◽  
Vol 75 (2) ◽  
pp. 313-320 ◽  
Author(s):  
Costantino Delizia ◽  
Primoz Moravec ◽  
Chiara Nicotera
Keyword(s):  
Finite Groups ◽  
Transitive Group ◽  
Transitive Relation ◽  
Transitive Groups

Finite groups in which a given property of two-generator subgroups is a transitive relation are investigated. We obtain a description of such groups and prove in particular that every finite soluble-transitive group is soluble. A classification of finite nilpotent-transitive groups is also obtained.

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.

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