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.

A note on Frobenius–Wielandt groups

2019 ◽  
Vol 22 (4) ◽  
pp. 637-645
Author(s):  
Gil Kaplan
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Frobenius Group ◽  
Point Of View ◽  
Coset Space ◽  
Frobenius Groups ◽  
Frobenius Theorem ◽  
Frobenius Kernel ◽  
Frobenius Complement ◽  
A Finite Group

AbstractLet G be a finite group. G is called a Frobenius–Wielandt group if there exists {H<G} such that {U=\langle H\cap H^{g}\mid g\in G-H\rangle} is a proper subgroup of H. The Wielandt theorem [H. Wielandt, Über die Existenz von Normalteilern in endlichen Gruppen, Math. Nachr. 18 1958, 274–280; Mathematische Werke Vol. 1, 769–775] on the structure of G generalizes the celebrated Frobenius theorem. From a permutation group point of view, considering the action of G on the coset space {G/H}, it states in particular that the subgroup {D=D_{G}(H)} generated by all derangements (fixed-point-free elements) is a proper subgroup of G. Let {W=U^{G}}, the normal closure of U in G. Then W is the subgroup generated by all elements fixing at least two points. We present the proof of the Wielandt theorem in a new way (Theorem 1.6, Corollary 1.7, Theorem 1.8) such that the unique component whose proof is not elementary or by the Frobenius theorem is the equality {W\cap H=U}. This presentation shows what can be achieved by elementary arguments and how Frobenius groups are involved in one case of Frobenius–Wielandt groups. To be more precise, Theorem 1.6 shows that there are two possible cases for a Frobenius–Wielandt group G with {H<G}: (a) {W=D} and {G=HW}, or (b) {W<D} and {HW<G}. In the latter case, {G/W} is a Frobenius group with a Frobenius complement {HW/W} and Frobenius kernel {D/W}.

Finite Groups with Four Relative Commutativity Degrees

2015 ◽  
Vol 22 (03) ◽  
pp. 449-458 ◽  
Author(s):  
A. Erfanian ◽  
M. Farrokhi D.G.
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Frobenius Group ◽  
Maximal Subgroups ◽  
Frobenius Kernel ◽  
A Finite Group

It is shown that a finite group G has four relative commutativity degrees if and only if G/Z(G) is a p-group of order p3 and G has no abelian maximal subgroups, or G/Z(G) is a Frobenius group with Frobenius kernel and complement isomorphic to ℤp × ℤp and ℤq, respectively, and the Sylow p-subgroup of G is abelian, where p and q are distinct primes.

scholarly journals On Periodic Groups Saturated with Finite Frobenius Groups

2021 ◽  
Vol 35 ◽  
pp. 73-86
Author(s):  
B. E. Durakov ◽  
◽  
A. I. Sozutov ◽  
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Frobenius Group ◽  
Finite Subgroup ◽  
Frobenius Groups ◽  
Locally Finite ◽  
Periodic Groups ◽  
Group A ◽  
Quotient Groups ◽  
Combinatorial Group

A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

A note on modular Frobenius groups

2020 ◽  
pp. 2250020
Author(s):  
Huiqin Cao ◽  
Jiwen Zeng
Keyword(s):  
Frobenius Group ◽  
Frobenius Groups

It is well known that Frobenius groups can be defined by their complement subgroups. But until now we cannot use a complement subgroup to define a modular Frobenius group. In the present paper, a generalization of Frobenius complements is used as a characterization of a class of modular Frobenius groups. In fact, we build a connection between modular Frobenius groups and Frobenius–Wielandt groups.

scholarly journals RESOLVABLE MENDELSOHN DESIGNS AND FINITE FROBENIUS GROUPS

2018 ◽  
Vol 98 (1) ◽  
pp. 1-13
Author(s):  
D. F. HSU ◽  
SANMING ZHOU
Keyword(s):  
Frobenius Group ◽  
Prime Factor ◽  
Frobenius Groups ◽  
Group Of Automorphisms ◽  
Euler’S Totient Function ◽  
Mendelsohn Design

We prove the existence and give constructions of a $(p(k)-1)$-fold perfect resolvable $(v,k,1)$-Mendelsohn design for any integers $v>k\geq 2$ with $v\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}\,k$ such that there exists a finite Frobenius group whose kernel $K$ has order $v$ and whose complement contains an element $\unicode[STIX]{x1D719}$ of order $k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K\rtimes \langle \unicode[STIX]{x1D719}\rangle$ as a group of automorphisms and is perfect when $k$ is a prime. As an application we prove that for any integer $v=p_{1}^{e_{1}}\cdots p_{t}^{e_{t}}\geq 3$ in prime factorisation and any prime $k$ dividing $p_{i}^{e_{i}}-1$ for $1\leq i\leq t$, there exists a resolvable perfect $(v,k,1)$-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if $k$ is even and divides $p_{i}-1$ for $1\leq i\leq t$, then there are at least $\unicode[STIX]{x1D711}(k)^{t}$ resolvable $(v,k,1)$-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where $\unicode[STIX]{x1D711}$ is Euler’s totient function.

scholarly journals FROBENIUS CIRCULANT GRAPHS OF VALENCY FOUR

2008 ◽  
Vol 85 (2) ◽  
pp. 269-282 ◽  
Author(s):  
ALISON THOMSON ◽  
SANMING ZHOU
Keyword(s):  
Cayley Graph ◽  
Cyclic Group ◽  
Interconnection Networks ◽  
Frobenius Group ◽  
Optimal Routing ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Frobenius Kernel ◽  
Even Order

AbstractA first kind Frobenius graph is a Cayley graph Cay(K,S) on the Frobenius kernel of a Frobenius group $K \rtimes H$ such that S=aH for some a∈K with 〈aH〉=K, where H is of even order or a is an involution. It is known that such graphs admit ‘perfect’ routing and gossiping schemes. A circulant graph is a Cayley graph on a cyclic group of order at least three. Since circulant graphs are widely used as models for interconnection networks, it is thus highly desirable to characterize those which are Frobenius of the first kind. In this paper we first give such a characterization for connected 4-valent circulant graphs, and then describe optimal routing and gossiping schemes for those which are first kind Frobenius graphs. Examples of such graphs include the 4-valent circulant graph with a given diameter and maximum possible order.

Full Frobenius Groups of Finite Morley Rank and the Feit-Thompson Theorem

2001 ◽  
Vol 7 (3) ◽  
pp. 315-328 ◽  
Author(s):  
Eric Jaligot
Keyword(s):  
Frobenius Group ◽  
Major Obstacle ◽  
Simple Groups ◽  
Morley Rank ◽  
Frobenius Groups ◽  
Finite Morley Rank

AbstractWe show how the notion of full Frobenius group of finite Morley rank generalizes that of bad group, and how it seems to be more appropriate when we consider the possible existence (still unknown) of nonalgebraic simple groups of finite Morley rank of a certain type, notably with no involution. We also show how these groups appear as a major obstacle in the analysis of FT-groups, if one tries to extend the Feit-Thompson theorem to groups of finite Morley rank.

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 THE CLASSIFICATION OF SOME MODULAR FROBENIUS GROUPS

2011 ◽  
Vol 85 (1) ◽  
pp. 11-18 ◽  
Author(s):  
JUANJUAN FAN ◽  
NI DU ◽  
JIWEN ZENG
Keyword(s):  
Normal Subgroup ◽  
Prime Number ◽  
Frobenius Group ◽  
Minimal Normal Subgroup ◽  
Complete Classification ◽  
Conjugacy Classes ◽  
Frobenius Groups

AbstractFix a prime number p. Let G be a p-modular Frobenius group with kernel N which is the minimal normal subgroup of G. We give the complete classification of G when N has three, four or five p-regular conjugacy classes. We also determine the structure of G when N has more than five p-regular conjugacy classes.

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.

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