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.

scholarly journals Transversals in permutation groups and factorisations of complete graphs

2002 ◽  
Vol 65 (2) ◽  
pp. 277-288 ◽  
Author(s):  
Gil Kaplan ◽  
Arieh Lev
Keyword(s):  
Permutation Group ◽  
Directed Graph ◽  
Sufficient Conditions ◽  
Combinatorial Problems ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Complete Graphs ◽  
Frobenius Theorem ◽  
Disjoint Cycles ◽  
Finite Set

Let G be a transitive permutation group acting on a finite set of order n. We discuss certain types of transversals for a point stabiliser A in G: free transversals and global transversals. We give sufficient conditions for the existence of such transversals, and show the connection between these transversals and combinatorial problems of decomposing the complete directed graph into edge disjoint cycles. In particular, we classify all the inner-transitive Oberwolfach factorisations of the complete directed graph. We mention also a connection to Frobenius theorem.

scholarly journals Wreath decompositions of finite permutation groups

1989 ◽  
Vol 40 (2) ◽  
pp. 255-279 ◽  
Author(s):  
L. G. Kovács
Keyword(s):  
Permutation Group ◽  
Wreath Product ◽  
Transitive Group ◽  
Wreath Products ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Formal Definition ◽  
The Third ◽  
Product Action ◽  
Factor Decomposition

There is a familiar construction with two finite, transitive permutation groups as input and a finite, transitive permutation group, called their wreath product, as output. The corresponding ‘imprimitive wreath decomposition’ concept is the first subject of this paper. A formal definition is adopted and an overview obtained for all such decompositions of any given finite, transitive group. The result may be heuristically expressed as follows, exploiting the associative nature of the construction. Each finite transitive permutation group may be written, essentially uniquely, as the wreath product of a sequence of wreath-indecomposable groups, amid the two-factor wreath decompositions of the group are precisely those which one obtains by bracketing this many-factor decomposition.If both input groups are nontrivial, the output above is always imprimitive. A similar construction gives a primitive output, called the wreath product in product action, provided the first input group is primitive and not regular. The second subject of the paper is the ‘product action wreath decomposition’ concept dual to this. An analogue of the result stated above is established for primitive groups with nonabelian socle.Given a primitive subgroup G with non-regular socle in some symmetric group S, how many subgroups W of S which contain G and have the same socle, are wreath products in product action? The third part of the paper outlines an algorithm which reduces this count to questions about permutation groups whose degrees are very much smaller than that of G.

scholarly journals On doubly transitive permutation groups of degree prime squared plus one

1977 ◽  
Vol 23 (2) ◽  
pp. 202-206 ◽  
Author(s):  
David Chillag
Keyword(s):  
Permutation Group ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Normal Extension ◽  
Stabilizer Of A Point

AbstractA doubly transitive permutation group of degreep2+ 1, pa prime, is proved to be doubly primitive forp≠ 2. We also show that if such a group is not triply transitive then either it is a normal extension ofP S L(2,p2) or the stabilizer of a point is a rank 3 group.

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 On the minimal degree of a transitive permutation group with stabilizer a 2-group

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Primož Potočnik ◽  
Pablo Spiga
Keyword(s):  
Fixed Points ◽  
Permutation Group ◽  
Minimal Degree ◽  
Simple Groups ◽  
Transitive Permutation Group ◽  
Finite Simple Groups ◽  
Trivial Element ◽  
Stabilizer Of A Point

AbstractThe minimal degree of a permutation group 𝐺 is defined as the minimal number of non-fixed points of a non-trivial element of 𝐺. In this paper, we show that if 𝐺 is a transitive permutation group of degree 𝑛 having no non-trivial normal 2-subgroups such that the stabilizer of a point is a 2-group, then the minimal degree of 𝐺 is at least \frac{2}{3}n. The proof depends on the classification of finite simple groups.

scholarly journals Normal subgroups of finite multiply transitive permutation groups

1974 ◽  
Vol 53 ◽  
pp. 103-107 ◽  
Author(s):  
Eiichi Bannai
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Normal Subgroups ◽  
Multiply Transitive

Wagner [5] and Ito [2] proved the following theorems respectively.THEOREM OF WAGNER. Let G be a triply transitive permutation group on a set Ω = {1,2, …, n}, and let n be odd and n > 4. If H is a normal subgroup (≠1) of G, then H is also triply transitive on Ω.

A theorem on transitive groups

1933 ◽  
Vol 29 (2) ◽  
pp. 257-259
Author(s):  
Garrett Birkhoff
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Transitive Permutation Group ◽  
Transitive Groups

Let be any transitive permutation group on the n symbols 1, …, n. Let be the subgroup of whose elements leave i fixed. Let ′ be the normalizer of , i.e., the subgroup of the symmetric group on 1, …, n transforming into itself. Let G′, G′1, G′2, etc., denote elements of ′. Finally, let ″ be the centralizer of , i.e., the subgroup in transforming every element of into itself.

scholarly journals Seminormal and subnormal subgroup lattices for transitive permutation groups

2006 ◽  
Vol 80 (1) ◽  
pp. 45-64
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Subnormal Subgroup ◽  
Subgroup Lattice ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Transitive Groups ◽  
Lattices Of Subgroups ◽  
Subgroup Lattices ◽  
A Chain ◽  
Seminormal Subgroup ◽  
Composition Factors

AbstractVarious lattices of subgroups of a finite transitive permutation group G can be used to define a set of ‘basic’ permutation groups associated with G that are analogues of composition factors for abstract finite groups. In particular G can be embedded in an iterated wreath product of a chain of its associated basic permutation groups. The basic permutation groups corresponding to the lattice L of all subgroups of G containing a given point stabiliser are a set of primitive permutation groups. We introduce two new subgroup lattices contained in L, called the seminormal subgroup lattice and the subnormal subgroup lattice. For these lattices the basic permutation groups are quasiprimitive and innately transitive groups, respectively.

scholarly journals Cycle Index, Weight Enumerator, and Tutte Polynomial

10.37236/1663 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Peter J. Cameron
Keyword(s):  
Permutation Group ◽  
Linear Code ◽  
Tutte Polynomial ◽  
Weight Enumerator ◽  
Permutation Groups ◽  
Cycle Index ◽  
Transitive Groups ◽  
Index Weight

With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to a trivial normalisation). There is a class of permutation groups (the IBIS groups) which includes the groups obtained from codes as above. With every IBIS group is associated a matroid; in the case of a group from a code, the matroid differs only trivially from that which arises directly from the code. In this case, the Tutte polynomial of the code specialises to the weight enumerator (by Greene's Theorem), and hence also to the cycle index. However, in another subclass of IBIS groups, the base-transitive groups, the Tutte polynomial can be derived from the cycle index but not vice versa. I propose a polynomial for IBIS groups which generalises both Tutte polynomial and cycle index.

On transitive permutation groups with a subgroup satisfying a certain conjugacy condition

1984 ◽  
Vol 36 (1) ◽  
pp. 69-86 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Permutation Group ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Pronormal Subgroup ◽  
Natural Representation ◽  
Conjugacy Condition ◽  
Sylow Subgroups ◽  
Point Stabilizer

AbstractLet G be transitive permutation group of degree n and let K be a nontrivial pronormal subgroup of G (that is, for all g in G, K and Kg are conjugate in (K, Kg)). It is shown that K can fix at most ½(n – 1) points. Moreover if K fixes exactly ½(n – 1) points then G is either An or Sn, or GL(d, 2) in its natural representation where n = 2d-1 ≥ 7. Connections with a result of Michael O'Nan are dicussed, and an application to the Sylow subgroups of a one point stabilizer is given.

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