Alexander Bors
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View article: Finite 2-groups with exactly three automorphism orbits
Finite 2-groups with exactly three automorphism orbits Open
We give a complete classification of the finite 2-groups 𝐺 for which the automorphism group Aut ( G ) \operatorname{Aut}(G) acting naturally on 𝐺 has three orbits. There are two infinite families and one additional group, of order …
View article: Wreath products and cascaded FSRs
Wreath products and cascaded FSRs Open
We show that the transition function of the cascaded connection of two FSRs can be viewed as a wreath product element. This allows us to study periods of cascaded connections with algebraic methods, obtaining both a general, nontrivial upp…
View article: Functional graphs of generalized cyclotomic mappings of finite fields
Functional graphs of generalized cyclotomic mappings of finite fields Open
The functional graph of a function $g:X\rightarrow X$ is the directed graph with vertex set $X$ the edges of which are of the form $x\rightarrow g(x)$ for $x\in X$. Functional graphs are heavily studied because they allow one to understand…
View article: Compositions and parities of complete mappings and of orthomorphisms
Compositions and parities of complete mappings and of orthomorphisms Open
We determine the permutation groups $P_{\mathrm{comp}}(\mathbb{F}_q),P_{\mathrm{orth}}(\mathbb{F}_q)\leq\operatorname{Sym}(\mathbb{F}_q)$ generated by the complete mappings, respectively the orthomorphisms, of the finite field $\mathbb{F}_…
View article: Generalized Cyclotomic Mappings: Switching Between Polynomial, Cyclotomic, and Wreath Product Form
Generalized Cyclotomic Mappings: Switching Between Polynomial, Cyclotomic, and Wreath Product Form Open
This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree with a suitable monomial function $x\mapsto ax^r$ on ea…
View article: Coset-wise affine functions and cycle types of complete mappings
Coset-wise affine functions and cycle types of complete mappings Open
Let $K$ be a finite field of characteristic $p$. We study a certain class of functions $K\rightarrow K$ that agree with an $\mathbb{F}_p$-affine function $K\rightarrow K$ on each coset of a given additive subgroup $W$ of $K$ - we call them…
View article: Words, permutations, and the nonsolvable length of a finite group
Words, permutations, and the nonsolvable length of a finite group Open
We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let w be a nontrivial word in d distinct variables and let G be a finite group for which the word map w_G\colon G^…
View article: Cycle types of complete mappings of finite fields
Cycle types of complete mappings of finite fields Open
We derive several existence results concerning cycle types and, more generally, the "mapping behavior" of complete mappings. Our focus is on so-called first-order cyclotomic mappings, which are functions on a finite field $\mathbb{F}_q$ th…
View article: Generalized cyclotomic mappings: Switching between polynomial, cyclotomic, and wreath product form
Generalized cyclotomic mappings: Switching between polynomial, cyclotomic, and wreath product form Open
This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree with a suitable monomial function $x\mapsto ax^r$ on ea…
View article: Finite $2$-groups with exactly three automorphism orbits
Finite $2$-groups with exactly three automorphism orbits Open
We give a complete classification of the finite $2$-groups $G$ for which the automorphism group $\operatorname{Aut}(G)$ acting naturally on $G$ has three orbits. There are two infinite families and one additional group, of order $2^9$. All…
View article: Worst-case approximability of functions on finite groups by endomorphisms and affine maps
Worst-case approximability of functions on finite groups by endomorphisms and affine maps Open
We study the maximum Hamming distance (or rather, the complementary notion of “minimum approximability”) of a general function on a finite group [Formula: see text] to either of the sets End(G) and Aff(G), of group endomorphisms of [Formul…
View article: Finite groups with an affine map of large order
Finite groups with an affine map of large order Open
Let $G$ be a group. A function $G\rightarrow G$ of the form $x\mapsto x^αg$ for a fixed automorphism $α$ of $G$ and a fixed $g\in G$ is called an affine map of $G$. In this paper, we study finite groups $G$ with an affine map of large orde…
View article: Finite transitive permutation groups with only small normaliser orbits
Finite transitive permutation groups with only small normaliser orbits Open
We study finite transitive permutation groups $G\leqslant\operatorname{Sym}(Ω)$ such that all orbits of the conjugation action on $G$ of the normaliser of $G$ in $\operatorname{Sym}(Ω)$ have size bounded by some constant. Our results exten…
View article: Finite groups with only small automorphism orbits
Finite groups with only small automorphism orbits Open
We study finite groups G such that the maximum length of an orbit of the natural action of the automorphism group Aut ( G ) {\mathrm{Aut}(G)} on G is bounded from above by a constant. Our main results are the following: Firstly, a …
View article: Documentation for the GAP code file OrbOrd.txt
Documentation for the GAP code file OrbOrd.txt Open
We give a comprehensive description of the functions and variables defined in the authors' GAP code file OrbOrd.txt, which serve mainly to compute (bounds on) the number of $\operatorname{Aut}(S)$-orbits on $S$, or the set or number of ele…
View article: Automorphism orbits and element orders in finite groups: almost-solubility and the Monster
Automorphism orbits and element orders in finite groups: almost-solubility and the Monster Open
For a finite group $G$, we denote by $ω(G)$ the number of $\operatorname{Aut}(G)$-orbits on $G$, and by $\operatorname{o}(G)$ the number of distinct element orders in $G$. In this paper, we are primarily concerned with the two quantities $…
View article: Formations of finite groups with the M. Hall property
Formations of finite groups with the M. Hall property Open
The first examples of formations which are arboreous (and therefore Hall) but not freely indexed (and therefore not locally extensible) are found. Likewise, the first examples of solvable formations which are freely indexed and arboreous (…
View article: Finite groups with an automorphism inverting, squaring or cubing a non-negligible fraction of elements
Finite groups with an automorphism inverting, squaring or cubing a non-negligible fraction of elements Open
Finite groups with an automorphism mapping a sufficiently large proportion of elements to their inverses, squares and cubes have been studied for a long time, and the gist of the results on them is that they are “close to being abelian”. I…
View article: Worst-case approximability of functions on finite groups by\n endomorphisms and affine maps
Worst-case approximability of functions on finite groups by\n endomorphisms and affine maps Open
We study the maximum Hamming distance (or rather, the complementary notion of\n"minimum approximability") of a general function on a finite group $G$ to\neither of the sets $\\operatorname{End}(G)$ and $\\operatorname{Aff}(G)$, of\ngroup e…
View article: Approximability of word maps by homomorphisms
Approximability of word maps by homomorphisms Open
Generalizing a recent result of Mann, we show that there is an explicit function $f:\left(0,1\right]\rightarrow\left(0,1\right]$ such that for every reduced word $w$, say in $d$ variables, there is an explicit reduced word $v$ in at most $…
View article: Fibers of automorphic word maps and an application to composition factors
Fibers of automorphic word maps and an application to composition factors Open
In this paper, we study the fibers of “automorphic word maps”, a certain generalization of word maps, on finite groups and on nonabelian finite simple groups in particular. As an application, we derive a structural restriction on finite gr…
View article: Nilpotency and the number of word maps of a finite group
Nilpotency and the number of word maps of a finite group Open
For a finite group $G$ and a non-negative integer $d$, denote by $Ω_d(G)$ the number of functions $G^d\rightarrow G$ that are induced by substitution into a word with variables among $X_1,\ldots,X_d$. In this note, we show that nilpotency …
View article: Fibers of word maps and the multiplicities of nonabelian composition factors
Fibers of word maps and the multiplicities of nonabelian composition factors Open
Call a reduced word $w$ multiplicity-bounding if and only if a finite group on which the word map of $w$ has a fiber of positive proportion $ρ$ can only contain each nonabelian finite simple group $S$ as a composition factor with multiplic…
View article: Finite groups with an automorphism of large order
Finite groups with an automorphism of large order Open
Let G be a finite group, and assume that G has an automorphism of order at least ρ | G | {\rho|G|} , with ρ ∈ ( 0 , 1 ) {\rho\in(0,1)} . We prove that if ρ > 1 / 2 {\rho>1/2} , then G is abelian, and if ρ > 1 / 10 …
View article: On the dynamics of endomorphisms of finite groups
On the dynamics of endomorphisms of finite groups Open
Aiming at a better understanding of finite groups as finite dynamical systems, we show that by a version of Fitting's Lemma for groups, each state space of an endomorphism of a finite group is a graph tensor product of a finite directed 1-…