scholarly journals A Class of Frobenius Groups

1959 ◽  
Vol 11 ◽  
pp. 39-47 ◽  
Author(s):  
Daniel Gorknstein
Keyword(s):  
Finite Field ◽  
Permutation Group ◽  
Finite Order ◽  
Frobenius Group ◽  
Primitive Element ◽  
Transitive Permutation Group ◽  
Affine Transformations ◽  
Frobenius Groups ◽  
One Dimensional ◽  
Cyclic Subgroups

If a group contains two subgroups A and B such that every element of the group is either in A or can be represented uniquely in the form aba', a, a’ in A, b ≠ 1 in B, we shall call the group an independent ABA-group. In this paper we shall investigate the structure of independent ABA -groups of finite order.A simple example of such a group is the group G of one-dimensional affine transformations over a finite field K. In fact, if we denote by a the transformation x’ = ωx, where ω is a primitive element of K, and by b the transformation x’ = —x + 1, it is easy to see that G is an independent ABA -group with respect to the cyclic subgroups A, B generated by a and b respectively.Since G admits a faithful representation on m letters (m = number of elements in K) as a transitive permutation group in which no permutation other than the identity leaves two letters fixed, and in which there is at least one permutation leaving exactly one letter fixed, G is an example of a Frobenius group.

On the Structure of Frobenius Groups

1957 ◽  
Vol 9 ◽  
pp. 587-596 ◽  
Author(s):  
Walter Feit
Keyword(s):  
Permutation Group ◽  
Frobenius Group ◽  
Faithful Representation ◽  
Transitive Permutation Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Frobenius Groups ◽  
Necessary And Sufficient

Let G be a group which has a faithful representation as a transitive permutation group on m letters in which no permutation other than the identity leaves two letters unaltered, and there is at least one permutation leaving exactly one letter fixed. It is easily seen that if G has order mh, a necessary and sufficient condition for G to have such a representation is that G contains a subgroup H of order h which is its own normalizer in G and is disjoint from all its conjugates. Such a group G is called a Frobenius group of type (h, m).

scholarly journals On the Existence of Frobenius Digraphical Representations

10.37236/7097 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Pablo Spiga
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Frobenius Group ◽  
Natural Extension ◽  
Partial Answer ◽  
Transitive Permutation Group ◽  
Frobenius Groups ◽  
Frobenius Complement ◽  
Vertex Set

A Frobenius group is a transitive permutation group that is not regular and such that only the identity fixes more than one point. A digraphical, respectively graphical, Frobenius representation, DFR and GFR for short, of a Frobenius group $F$ is a digraph, respectively graph, whose automorphism group as a group of permutations of the vertex set is $F$. The problem of classifying which Frobenius groups admit a DFR and GFR has been proposed by Mark Watkins and Thomas Tucker and is a natural extension of the problem of classifying which groups that have a digraphical, respectively graphical, regular representation.In this paper, we give a partial answer to a question of Mark Watkins and Thomas Tucker concerning Frobenius representations: "All but finitely many Frobenius groups with a given Frobenius complement have a DFR".  

Some generalized characters associated to a transitive permutation group

2020 ◽  
Vol 23 (3) ◽  
pp. 393-397
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Frobenius Group ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Transitive Permutation Group ◽  
Point Stabilizer ◽  
Necessary And Sufficient ◽  
Regular Character ◽  
Permutation Character

AbstractLet G be a finite transitive permutation group of degree n, with point stabilizer {H\neq 1} and permutation character π. For every positive integer t, we consider the generalized character {\psi_{t}=\rho_{G}-t(\pi-1_{G})}, where {\rho_{G}} is the regular character of G and {1_{G}} the 1-character. We give necessary and sufficient conditions on t (and G) which guarantee that {\psi_{t}} is a character of G. A necessary condition is that {t\leq\min\{n-1,\lvert H\rvert\}}, and it turns out that {\psi_{t}} is a character of G for {t=n-1} resp. {t=\lvert H\rvert} precisely when G is 2-transitive resp. a Frobenius group.

SL(2,5) and Frobenius Galois Groups Over Q

1980 ◽  
Vol 32 (2) ◽  
pp. 281-293 ◽  
Author(s):  
Jack Sonn
Keyword(s):  
Galois Group ◽  
Permutation Group ◽  
Number Field ◽  
Frobenius Group ◽  
Rational Numbers ◽  
Galois Groups ◽  
Transitive Permutation Group ◽  
Nonsolvable Group ◽  
Imaginary Field

A finite transitive permutation group G is called a Frobenius group if every element of G other than 1 leaves at most one letter fixed, and some element of G other than 1 leaves some letter fixed. It is proved in [20] (and sketched below) that if k is a number field such that SL(2, 5) and one other nonsolvable group Ŝ5 of order 240 are realizable as Galois groups over k, then every Frobenius group is realizable over k. It was also proved in [20] that there exists a quadratic (imaginary) field over which these two groups are realizable. In this paper we prove that they are realizable over the rationals Q, hence we ObtainTHEOREM 1. Every Frobenius group is realizable as the Galois group of an extension of the rational numbersQ.

Infinite Frobenius Groups Generated by Elements of Order 3

2020 ◽  
Vol 27 (04) ◽  
pp. 741-748
Author(s):  
Nanying Yang ◽  
Daria Victorovna Lytkina ◽  
Victor Danilovich Mazurov ◽  
Archil Khazeshovich Zhurtov
Keyword(s):  
Finite Order ◽  
Semidirect Product ◽  
Frobenius Group ◽  
Identity Element ◽  
Normal Structure ◽  
Product Formula ◽  
Frobenius Groups ◽  
Periodic Subgroup ◽  
Splitting Automorphism

A semidirect product [Formula: see text] of groups F and H is called a Frobenius group if the following two conditions are satisfied: (F1) H acts freely on F, that is, [Formula: see text] for f in F and h in H only if h = 1 or f = 1. (F2) Every non-identity element [Formula: see text] of finite order n induces in F by conjugation in G a splitting automorphism, that is, [Formula: see text] for every [Formula: see text]; in other words, the order of [Formula: see text] is equal to n. We describe the normal structure of a Frobenius group with periodic subgroup H generated by elements of order 3.

scholarly journals Frobenius groups of low rank

2021 ◽  
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Frobenius Group ◽  
Low Rank ◽  
Special Situation ◽  
Frobenius Groups ◽  
Frobenius Kernel

AbstractLet G be a finite Frobenius group of degree n. We show, by elementary means, that n is a power of some prime p provided the rank $${\mathrm{rk}}(G)\le 3+\sqrt{n+1}$$ rk ( G ) ≤ 3 + n + 1 . Then the Frobenius kernel of G agrees with the (unique) Sylow p-subgroup of G. So our result implies the celebrated theorems of Frobenius and Thompson in a special situation.

On a conjecture of Hughes

1967 ◽  
Vol 63 (3) ◽  
pp. 647-652 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Permutation Group ◽  
Translation Plane ◽  
Collineation Group ◽  
Affine Plane ◽  
Finite Projective Plane ◽  
Transitive Permutation Group ◽  
Condition C

D. R. Hughes stated the following conjecture: If π is a finite projective plane satisfying the condition: (C)π contains a collineation group δ inducing a doubly transitive permutation group δ* on the points of a line g, fixed under δ, then the corresponding affine plane πg is a translation plane.

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 Constructions for arc-transitive digraphs

1995 ◽  
Vol 59 (1) ◽  
pp. 61-80 ◽  
Author(s):  
Marston Conder ◽  
Peter Lorimer ◽  
Cheryl Praeger
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Finite Groups ◽  
Transitive Permutation Group ◽  
Group Of Automorphisms ◽  
Representations Of Finite Groups ◽  
Regular Digraph

AbstractA number of constructions are given for arc-transitive digraphs, based on modifications of permutation representations of finite groups. In particular, it is shown that for every positive integer s and for any transitive permutation group p of degree k, there are infinitely many examples of a finite k-regular digraph with a group of automorphisms acting transitively on s-arcs (but not on (s + 1)-arcs), such that the stabilizer of a vertex induces the action of P on the out-neighbour set.

On Unsolvable Groups of Degree p = 4q + 1, p and q Primes

1967 ◽  
Vol 19 ◽  
pp. 583-589 ◽  
Author(s):  
K. I. Appel ◽  
E. T. Parker
Keyword(s):  
Permutation Group ◽  
Alternating Group ◽  
Transitive Permutation Group ◽  
University Of Illinois ◽  
The University

This paper presents two results. They are:Theorem 1. Let G be a doubly transitive permutation group of degree nq + 1 where a is a prime and n < g. If G is neither alternating nor symmetric, then G has Sylow q-subgroup of order only q.Result 2. There is no unsolvable transitive permutation group of degree p = 29, 53, 149, 173, 269, 293, or 317 properly contained in the alternating group of degree p.Result 2 was demonstrated by a computation on the Illiac II computer at the University of Illinois.

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