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Alexander Bors

Publications and source records attributed to Alexander Bors.

At least 19 recordsLinked to original sources
arXiv2023

Wreath products and cascaded FSRs

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 upper bound on the maximum period of a cascaded connection and a complete, explicit understanding of the periods in the important case of the cascaded connection of an $n$-dimensional De Bruijn sequence into an $m$-dimensional linear FSR.

math.NT↗
arXiv2023

Functional graphs of generalized cyclotomic mappings of finite fields

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 the behavior of $g$ under iteration (i.e., to understand the discrete dynamical system $(X,g)$), which has various applications, especially when $X$ is a finite field $\mathbb{F}_q$. This paper is an extensive study of the functional graphs of so-called index $d$ generalized cyclotomic mappings of $\mathbb{F}_q$, which are a natural and manageable generalization of monomial functions. We provide both theoretical results on the structure of their functional graphs and Las Vegas algorithms for solving fundamental problems, such as parametrizing the connected components of the functional graph by representative vertices, or describing the structure of a connected component given by a representative vertex. The complexity of these algorithms is analyzed in detail, and we make the point that for fixed index $d$ and most prime powers $q$ (in the sense of asymptotic density), suitable implementations of these algorithms have an expected runtime that is polynomial in $\log{q}$ on quantum computers, whereas their expected runtime is subexponential in $\log{q}$ on a classical computer. We also discuss four special cases in which one can devise Las Vegas algorithms with this kind of complexity behavior over most finite fields that solve the graph isomorphism problem for functional graphs of generalized cyclotomic mappings.

math.NT↗
arXiv2022

Compositions and parities of complete mappings and of orthomorphisms

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 $\mathbb{F}_q$ -- both are equal to $\operatorname{Sym}(\mathbb{F}_q)$ unless $q\in\{2,3,4,5,8\}$. More generally, denote by $P_{\mathrm{comp}}(G)$, respectively $P_{\mathrm{orth}}(G)$, the subgroup of $\operatorname{Sym}(G)$ generated by the complete mappings, respectively the orthomorphisms, of the group $G$. Using recent results of Eberhard-Manners-Mrazovi\'c and M\"uyesser-Pokrovskiy, we show that for each large enough finite group $G$ that has a complete mapping (i.e., whose Sylow $2$-subgroups are trivial or noncyclic), $P_{\mathrm{comp}}(G)=\operatorname{Sym}(G)$ and $P_{\mathrm{orth}}(G)\geq\operatorname{Alt}(G)$. We also prove that $P_{\mathrm{orth}}(G)=\operatorname{Sym}(G)$ for every large enough finite solvable group $G$ that has a complete mapping. Proving these results requires us to study the parities of complete mappings and of orthomorphisms. Some connections with known results in cryptography and with parity types of Latin squares are also discussed.

math.GR↗
arXiv2021

Coset-wise affine functions and cycle types of complete mappings

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 $W$-coset-wise $\mathbb{F}_p$-affine functions of $K$. We show that these functions form a permutation group on $K$ with the structure of an imprimitive wreath product and characterize which of them are complete mappings of $K$. As a consequence, we are able to provide various new examples of cycle types of complete mappings of $K$, including that $K$ has a complete mapping moving all elements of $K$ in one cycle if $p>2$.

math.NT↗
arXiv2021

Finite groups with an affine map of large order

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 order. More precisely, we show that if $G$ admits an affine map of order larger than $\frac{1}{2}|G|$, then $G$ is solvable of derived length at most $3$. We also show that more generally, for each $ρ\in\left(0,1\right]$, if $G$ admits an affine map of order at least $ρ|G|$, then the largest solvable normal subgroup of $G$ has derived length at most $4\lfloor\log_2(ρ^{-1})\rfloor+3$.

math.GR↗
arXiv2021

Cycle types of complete mappings of finite fields

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 $\mathbb{F}_q$ that fix $0$ and restrict to the multiplication $x\mapsto a_ix$ by a fixed element $a_i\in\mathbb{F}_q$ on each coset $C_i$ of a given subgroup $C$ of $\mathbb{F}_q^{\ast}$. The gist of two of our main results is that as long as $q$ is large enough relative to the index $|\mathbb{F}_q^{\ast}:C|$, all cycle types of first-order cyclotomic permutations with only long cycles on $\mathbb{F}_q^{\ast}$ can be achieved through a complete mapping, as can all permutations of the cosets of $C$. Our third main result provides new examples of complete mappings $f$ such that both $f$ and its associated orthomorphism $f+\operatorname{id}$ permute the nonzero field elements in one cycle.

math.NT↗
arXiv2021

Generalized cyclotomic mappings: Switching between polynomial, cyclotomic, and wreath product form

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 each coset of the index $d$ subgroup of $\mathbb{F}_q^{\ast}$. We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications thereof that concern index $d$ generalized cyclotomic permutations of $\mathbb{F}_q$ and pertain to cycle structures, the classification of $(q-1)$-cycles and involutions, as well as inversion.

math.NT↗
arXiv2021

Orbits of Sylow subgroups of finite permutation groups

We say that a finite group $G$ acting on a set $Ω$ has Property $(*)_p$ for a prime $p$ if $P_ω$ is a Sylow $p$-subgroup of $G_ω$ for all $ω\inΩ$ and Sylow $p$-subgroups $P$ of $G$. Property $(*)_p$ arose in the recent work of Tornier (2018) on local Sylow $p$-subgroups of Burger-Mozes groups, and he determined the values of $p$ for which the alternating group $A_n$ and symmetric group $S_n$ acting on $n$ points has Property $(*)_p$. In this paper, we extend this result to finite $2$-transitive groups and we give a structural characterisation result for the finite primitive groups that satisfy Property $(*)_p$ for an allowable prime $p$.

math.GR↗
arXiv2020

Finite $2$-groups with exactly three automorphism orbits

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 of them are Suzuki $2$-groups, and they appear in an earlier classification of Dornhoff.

math.GR↗
arXiv2020

Automorphism orbits and element orders in finite groups: almost-solubility and the Monster

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 $\mathfrak{d}(G):=ω(G)-\operatorname{o}(G)$ and $\mathfrak{q}(G):=ω(G)/\operatorname{o}(G)$, each of which may be viewed as a measure for how far $G$ is from being an AT-group in the sense of Zhang (that is, a group with $ω(G)=\operatorname{o}(G)$). We show that the index $|G:\operatorname{Rad}(G)|$ of the soluble radical $\operatorname{Rad}(G)$ of $G$ can be bounded from above both by a function in $\mathfrak{d}(G)$ and by a function in $\mathfrak{q}(G)$ and $\operatorname{o}(\operatorname{Rad}(G))$. We also obtain a curious quantitative characterisation of the Fischer-Griess Monster group $\operatorname{M}$.

math.GR↗
arXiv2020

Finite transitive permutation groups with only small normaliser orbits

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 extend recent results, due to the first author, on finite abstract groups $G$ such that all orbits of the natural action of the automorphism group $\operatorname{Aut}(G)$ on $G$ have size bounded by some constant.

math.GR↗
arXiv2019

Computation of orders and cycle lengths of automorphisms of finite solvable groups

Let $G$ be a finite solvable group, given through a refined consistent polycyclic presentation, and $α$ an automorphism of $G$, given through its images of the generators of $G$. In this paper, we discuss algorithms for computing the order of $α$ as well as the cycle length of a given element of $G$ under $α$. We give correctness proofs and discuss the theoretical complexity of these algorithms. Along the way, we carry out detailed complexity analyses of several classical algorithms on finite polycyclic groups.

math.GR↗
arXiv2019

Worst-case approximability of functions on finite groups by endomorphisms and affine maps

We study the maximum Hamming distance (or rather, the complementary notion of "minimum approximability") of a general function on a finite group $G$ to either of the sets $\operatorname{End}(G)$ and $\operatorname{Aff}(G)$, of group endomorphisms of $G$ and affine maps on $G$ respectively, the latter being a certain generalization of endomorphisms. We give general bounds on these two quantities and discuss an infinite class of extremal examples (where each of the two Hamming distances can be made as large as generally possible). Finally, we compute the precise values of the two quantities for all finite groups $G$ with $|G|\leq15$.

math.GR↗
arXiv2019

Documentation for the GAP code file OrbOrd.txt

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 element orders in $S$ for nonabelian finite simple groups of Lie type $S$.

math.GR↗
arXiv2019

Finite groups with only small automorphism orbits

We study finite groups $G$ such that the maximum length of an orbit of the natural action of the automorphism group $\operatorname{Aut}(G)$ on $G$ is bounded from above by a constant. Our main results are the following: Firstly, a finite group $G$ only admits $\operatorname{Aut}(G)$-orbits of length at most $3$ if and only if $G$ is cyclic of one of the orders $1$, $2$, $3$, $4$ or $6$, or $G$ is the Klein four group or the symmetric group of degree $3$. Secondly, there are infinitely many finite ($2$-)groups $G$ such that the maximum length of an $\operatorname{Aut}(G)$-orbit on $G$ is $8$. Thirdly, the order of a $d$-generated finite group $G$ such that $G$ only admits $\operatorname{Aut}(G)$-orbits of length at most $c$ is explicitly bounded from above in terms of $c$ and $d$. Fourthly, a finite group $G$ such that all $\operatorname{Aut}(G)$-orbits on $G$ are of length at most $23$ is solvable.

math.GR↗
arXiv2019

Words, permutations, and the nonsolvable length of a finite group

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 $w_G:G^d\rightarrow G$ has a fiber of size at least $ρ|G|^d$ for some fixed $ρ>0$. We show that, for certain words $w$, this implies that $G$ has a normal solvable subgroup of index bounded above in terms of $w$ and $ρ$. We also show that, for a larger family of words $w$, this implies that the nonsolvable length of $G$ is bounded above in terms of $w$ and $ρ$, thus providing evidence in favor of a conjecture of Larsen. Along the way we obtain results of some independent interest, showing roughly that most elements of large finite permutation groups have large support.

math.GR↗
arXiv2018

Formations of finite groups with the M. Hall property

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 (and therefore Hall) but not locally extensible are constructed. Some open questions are also mentioned.

math.GR↗