arXiv subjects

Alexander Bors

Publications and source records attributed to Alexander Bors.

32 records · Page 2Linked to original sources

arXiv2018

Finite groups with a large automorphism orbit

We study the nonabelian composition factors of a finite group $G$ assumed to admit an $\operatorname{Aut}(G)$-orbit of length at least $ρ|G|$, for a given $ρ\in\left(0,1\right]$. Our main results are the following: The orders of the nonabelian composition factors of $G$ are then bounded in terms of $ρ$, and if $ρ>\frac{18}{19}$, then $G$ is solvable. On the other hand, for each nonabelian finite simple group $S$, there is a constant $c(S)\in\left(0,1\right]$ such that $S$ occurs with arbitrarily large multiplicity as a composition factor in some finite group $G$ having an $\operatorname{Aut}(G)$-orbit of length at least $c(S)|G|$.

math.GR↗

arXiv2017

Approximability of word maps by homomorphisms

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 $3d$ variables (nontrivial if the length of $w$ is at least $2$) such that for all $ρ\in\left(0,1\right]$, the following holds: If $G$ is any finite group for which the word map $w_G:G^d\rightarrow G$ agrees with some fixed homomorphism $G^d\rightarrow G$ on at least $ρ|G|^d$ many arguments, then the word map $v_G:G^{3d}\rightarrow G$ has a fiber of size at least $f(ρ)|G|^{3d}$. We also discuss some applications of this result.

math.GR↗

arXiv2017

Nilpotency and the number of word maps of a finite group

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 of $G$ can be characterized through the asymptotic growth rate of $Ω_d(G)$ as $d\to\infty$.

math.GR↗

arXiv2017

Fibers of word maps and the multiplicities of nonabelian composition factors

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 multiplicity bounded in terms of $ρ$ and $S$. In this paper, based on recent work of Nikolov, we present methods to show that a given reduced word is multiplicity-bounding and apply them to give some nontrivial examples of multiplicity-bounding words, such as words of the form $x^e$, where $x$ is a single variable and $e$ an odd integer.

math.GR↗

arXiv2016

Fibers of automorphic word maps and an application to composition factors

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 groups $G$ where, for some fixed nonempty reduced word $w$ in $d$ variables and some fixed $ρ\in\left(0,1\right]$, the word map $w_G$ on $G$ has a fiber of size at least $ρ|G|^d$: No sufficiently large alternating group and no (classical) simple group of Lie type of sufficiently high rank can occur as a composition factor of such a group $G$.

math.GR↗

arXiv2016

Finite groups with an automorphism inverting, squaring or cubing a non-negligible fraction of elements

There are various results in the literature which are part of the general philosophy that a finite group for which a certain parameter (for example, the number of conjugacy classes or the maximum number of elements inverted, squared or cubed by a single automorphism) is large enough must be close to being abelian. In this paper, we show the following: Fix a real number $ρ$ with $0<ρ\leq 1$. Then a finite group $G$ with an automorphism inverting or squaring at least $ρ|G|$ of the elements in $G$ is "almost abelian" in the sense that both the index and the derived length of the solvable radical of $G$ are bounded. Furthermore, if $G$ has an automorphism cubing at least $ρ|G|$ of the elements in $G$, then $G$ is "almost solvable" in the sense that the index of the solvable radical of $G$ is bounded.

math.GR↗

arXiv2016

A note on finite groups with an automorphism inverting or squaring a non-negligible fraction of elements

We show that for a finite group $G$, the commuting probability of $G$ can be explicitly bounded from below in a nontrivial way by a function in the maximum fraction of elements inverted resp. squared by an automorphism of $G$. Using these bounds together with a result of Guralnick and Robinson gives upper bounds on the index of the Fitting subgroup of $G$ under each of the two conditions that $G$ have an automorphism inverting resp. squaring at least $ρ|G|$ many elements in $G$, for $ρ\in\left(0,1\right]$ fixed. This is an improvement on previous results of the author.

math.GR↗

arXiv2015

A bound on element orders in the holomorph of a finite group

Let $G$ be a finite group. We prove a theorem implying that the orders of elements of the holomorph $\operatorname{Hol}(G)$ are bounded from above by $|G|$, and we discuss an application to bounding automorphism orders of finite groups.

math.GR↗

arXiv2015

Finite groups with an automorphism of large order

Let $G$ be a finite group, and assume that $G$ has an automorphism of order at least $ρ|G|$, with $ρ\in\left(0,1\right)$. Generalizing recent analogous results of the author on finite groups with a large automorphism cycle length, we prove that if $ρ>1/2$, then $G$ is abelian, and if $ρ>1/10$, then $G$ is solvable, whereas in general, the assumption implies $[G:\operatorname{Rad}(G)]\leqρ^{-1.78}$, where $\operatorname{Rad}(G)$ denotes the solvable radical of $G$. Furthermore, we generalize an example of Horoševski\uı to show that in finite groups, the quotient of the maximum automorphism order by the maximum automorphism cycle length may be arbitrarily large.

math.GR↗

arXiv2015

Cycle lengths in finite groups and the size of the solvable radical

We prove the following: For any $ρ\in\left(0,1\right)$, if a finite group $G$ has an automorphism with a cycle of length at least $ρ\cdot|G|$, then the index of the solvable radical $\operatorname{Rad}(G)$ in $G$ is bounded from above in terms of $ρ$, and such a condition is strong enough to imply solvability of $G$ if and only if $ρ>\frac{1}{10}$. Furthermore, considering, for exponents $e\in\left(0,1\right)$, the condition that a finite group $G$ have an automorphism with a cycle of length at least $|G|^e$, such a condition is strong enough to imply $|\operatorname{Rad}(G)|\to\infty$ for $|G|\to\infty$ if and only if $e>\frac{1}{3}$. We also prove similar results for a larger class of bijective self-transformations of finite groups, so-called periodic affine maps.

math.GR↗

arXiv2015

Classification of finite group automorphisms with a large cycle

Let $ψ$ be a permutation of a finite set $X$. We define $λ(ψ)$ to be the largest fraction of elements of $X$ lying on a single cycle of $ψ$. For a finite group $G$, we define $λ(G)$ to be the maximum among the values $λ(α)$, where $α$ runs through the automorphisms of $G$. In this paper, we develop tools to deal with questions related to $λ$-values of finite groups and of their automorphisms. As a consequence, we will be able to give a classification, up to a natural notion of isomorphism, of those pairs $(G,α)$ where $G$ is a finite group, $α$ is an automorphism of $G$ and $λ(α)\geq\frac{1}{2}$.

math.GR↗

arXiv2015

On finite groups where the order of every automorphism is a cycle length

Using Frobenius normal forms of matrices over finite fields as well as the Burnside Basis Theorem, we give a direct proof of Horoševskiĭ's result that every automorphism $α$ of a finite nilpotent group has a cycle whose length coincides with $\mathrm{ord}(α)$. Also, we give two new sufficient conditions for an automorphism $α$ of an arbitrary finite group to satisfy this property, namely when $\mathrm{ord}(α)$ is a product of at most two prime powers or when $α$ has a sufficiently large cycle. This will allow us to show that the least order of a group where this property is violated is 120. Finally, we observe that any finite group embeds both into a group with this property (as all finite symmetric groups enjoy the property) as well as into a finite group not having this property.

math.GR↗

arXiv2014

On the dynamics of endomorphisms of finite groups

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 $1$-tree whose cycle is a loop with a disjoint union of cycles, generalizing results of Hernández-Toledo on linear finite dynamical systems, and we fully characterize the possible forms of state spaces of nilpotent endomorphisms via their "ramification behavior". Finally, as an application, we will count the isomorphism types of state spaces of endomorphisms of finite cyclic groups in general, extending results of Hernández-Toledo on primary cyclic groups of odd order.

math.GR↗

arXiv2014

On the endomorphism monoids of some groups with abelian automorphism group

We investigate the endomorphism monoids of certain finite $p$-groups of order $p^8$ first studied by Jonah and Konvisser in 1975 as examples for finite $p$-groups with abelian automorphism group, and we show some necessary conditions for a finite $p$-group to have commutative endomorphism monoid. As a by-product, apart from formulas for the number of conjugacy classes of endomorphisms of said groups, we will be able to derive the following: There exist nonabelian groups with commutative endomorphism monoid, and having commutative endomorphism monoid is a group property strictly stronger than having abelian automorphism group. Furthermore, using a result of Curran, this will enable us to give, for all primes $p$, examples of finite $p$-groups which are direct products and have abelian automorphism group.

math.GR↗