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 action on Carter subgroups

2020 ◽  
Vol 30 (05) ◽  
pp. 1073-1080
Author(s):  
Güli̇n Ercan ◽  
İsmai̇l Ş. Güloğlu
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Semidirect Product ◽  
Frobenius Group ◽  
Product Formula ◽  
Finite Solvable Group ◽  
Carter Subgroup ◽  
Text Length ◽  
Fitting Height ◽  
A Finite Group

Let [Formula: see text] be a finite solvable group and [Formula: see text] be a subgroup of [Formula: see text]. Suppose that there exists an [Formula: see text]-invariant Carter subgroup [Formula: see text] of [Formula: see text] such that the semidirect product [Formula: see text] is a Frobenius group with kernel [Formula: see text] and complement [Formula: see text]. We prove that the terms of the Fitting series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the Fitting series of [Formula: see text], and the Fitting height of [Formula: see text] may exceed the Fitting height of [Formula: see text] by at most one. As a corollary it is shown that for any set of primes [Formula: see text], the terms of the [Formula: see text]-series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the [Formula: see text]-series of [Formula: see text], and the [Formula: see text]-length of [Formula: see text] may exceed the [Formula: see text]-length of [Formula: see text] by at most one. These theorems generalize the main results in [E. I. Khukhro, Fitting height of a finite group with a Frobenius group of automorphisms, J. Algebra 366 (2012) 1–11] obtained by Khukhro.

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.

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.

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.

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 semi-σ-nilpotent finite groups

2019 ◽  
Vol 18 (10) ◽  
pp. 1950200
Author(s):  
Chi Zhang ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Semidirect Product ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Product Formula ◽  
Nilpotent Subgroup ◽  
Chief Factor ◽  
Group A ◽  
A Finite Group

Let [Formula: see text] be a partition of the set [Formula: see text] of all primes and [Formula: see text] a finite group. A chief factor [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-central if the semidirect product [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. [Formula: see text] is called [Formula: see text]-nilpotent if every chief factor of [Formula: see text] is [Formula: see text]-central. We say that [Formula: see text] is semi-[Formula: see text]-nilpotent (respectively, weakly semi-[Formula: see text]-nilpotent) if the normalizer [Formula: see text] of every non-normal (respectively, every non-subnormal) [Formula: see text]-nilpotent subgroup [Formula: see text] of [Formula: see text] is [Formula: see text]-nilpotent. In this paper we determine the structure of finite semi-[Formula: see text]-nilpotent and weakly semi-[Formula: see text]-nilpotent groups.

scholarly journals ON THE PARTIAL DIFFERENCE SETS IN CAYLEY DERANGEMENT GRAPHS

2019 ◽  
pp. 651
Author(s):  
Modjtaba Ghorbani ◽  
Mina Rajabi-Parsa
Keyword(s):  
Finite Group ◽  
Frobenius Group ◽  
Identity Element ◽  
Difference Sets ◽  
Difference Set ◽  
Partial Difference Set ◽  
Partial Difference ◽  
Partial Difference Sets ◽  
A Finite Group

Let $G$ be a finite group. The set $D\subseteq G$with $|D|=k$ is called a $(n,k,\lambda,\mu)$-partial difference set(PDS) in $G$ if the differences $d_1d_2 ^{-1}, d_2,d_2\in  D, d_1\neq d_2$, represent each non-identity element in $D$ exactly $\lambda$  times and each non-identity element in $G-\{D\}$ exactly $\mu$  times.In the present paper, we determine for which group $G\in \{D_{2n},T_{4n},U_{6n},V_{8n}\}$ the derangement set is a PDS. We also prove that the derangement set of a Frobenius group is a PDS.

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).

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