arXiv subjects

Kıvanç Ersoy

Publications and source records attributed to Kıvanç Ersoy.

6 recordsLinked to original sources

arXiv2026

Brauer-Fowler Bounds for Elements of Odd Prime Order

The Brauer--Fowler theorem bounds the order of a finite simple group in terms of the order of the centralizer of an involution. Hartley proved an automorphism version, bounding the order of a finite simple group in terms of the order of an automorphism and the order of its fixed-point subgroup. Strunkov asked whether, in the involution case, the order of the centralizer can be replaced by the number of involutions commuting with the given involution; this was recently answered affirmatively by Skresanov. Motivated by this question, and in contrast with Hartley's use of the order of a fixed-point subgroup, we study what can be deduced from counting elements of prescribed prime order inside such a subgroup. We show that the direct analogue of Skresanov's result fails for elements of odd prime order. We prove that if $G$ is a finite simple group, $x\in G$ has odd prime order $p$, $C_G(x)$ contains at most $k$ elements of order $p$, and the exponent of $C_G(x)$ is at most $e$, then $|G|$ is bounded in terms of $k$ and $e$.

math.GR↗

arXiv2026

Mixed identities for simple locally finite groups

A mixed identity of a group is a nontrivial word with constants that vanishes under every substitution of its variables. We derive lower bounds for the length of mixed identities in finite simple groups of Lie type, and characterise exactly those families of such groups of bounded rank which satisfy mixed identities of bounded length. We classify the infinite simple locally finite groups admitting a mixed identity: apart from an explicit list of alternating, finitary linear classical, and non-simply-laced groups of Lie type, no such group exists. For the groups in this list, we determine when mixed identities must be singular and obtain restrictions on their critical constants. Moreover, we prove that simple compact Lie groups do not admit mixed identities.

math.GR↗

arXiv2026

Inverse ambiguous maps on infinite groups

Let $G$ be a group. A bijection $f\colon G\to G$ is called inverse ambiguous if $f^{-1}(x)=f(x)^{-1}$ for every $x\in G$. We prove that every infinite group admits an inverse ambiguous function. An inverse ambiguous automorphism can exist only on an abelian group. We classify the finitely generated and divisible abelian groups admitting such automorphisms. If $A\cong\Z^r\oplus T$, where $T$ is finite, then $A$ admits an inverse ambiguous automorphism if and only if $r$ is even and $T$ admits one. Together with Toborg's finite classification, this gives an explicit classification of the finitely generated abelian groups admitting such automorphisms. For a divisible abelian group $A\cong\mathbb Q^{(κ_0)}\oplus \bigoplus_p C_{p^\infty}^{(κ_p)}$, such an automorphism exists if and only if none of $κ_0$, $κ_2$, or $κ_p$ for $p\equiv3\pmod4$ is a finite odd cardinal. We also prove that if every proper subgroup of an infinite locally finite group $G$ admits an inverse ambiguous automorphism, then $G$ admits one that leaves every subgroup invariant. Finally, we prove that a residually finite group whose finite quotients admit inverse ambiguous automorphisms is abelian. However, we prove the existence of infinite residually finite abelian groups whose finite quotients all admit such automorphisms, but the group itself does not.

math.GR↗

arXiv2026

A residually finite analogue of Kegel's theorem on splitting automorphisms

Thompson proved that every finite group admitting a fixed-point-free automorphism of prime order is nilpotent, and Kegel showed that the same conclusion holds for finite groups admitting a splitting automorphism of prime order. Motivated by these results, Sozutov asked whether a $p'$-group admitting a splitting automorphism of prime order is locally nilpotent if \[ \langle g, g^φ, \dots, g^{φ^{p-1}} \rangle \] is nilpotent for every $g \in G$, \cite[Problem 10.59]{kourovka21}. We prove that if $G$ is a periodic residually finite group admitting a splitting automorphism of prime order $p$ then $G$ is nilpotent of class bounded in terms of $p$. This gives an affirmative answer, for residually finite groups, to the problem of Sozutov. We also prove that a possible counterexample to Sozutov's problem cannot be a Tarski monster.

math.GR↗

arXiv2026

On minimal non-sofic and $ω$-non-sofic groups

We investigate structural properties of non-sofic groups, assuming that such groups exist. We introduce and study two classes: minimal non-sofic groups and $ω$-non-sofic groups. For minimal non-sofic groups, we establish strong structural restrictions. In particular, we show that if $G$ is a minimal non-sofic group and $M$ is a finitely generated residually finite maximal normal subgroup of $G$, then $M$ is central and $G$ is a perfect central extension of a finitely generated non-amenable simple group. On the other hand, we show that locally graded non-sofic groups are necessarily $ω$-non-sofic. More precisely, such groups contain finitely generated non-sofic subgroups admitting strictly decreasing chains of finitely generated normal subgroups whose intersection is nontrivial and lies in the profinite residual. Finally, using results on existentially closed groups, we prove that the existence of a non-sofic group implies the existence of a countable existentially closed non-sofic group whose centralizers form a densely ordered chain of non-sofic subgroups of order type $(\mathbb{Q},\leq)$. In particular, we show that if a non-sofic group exists, then the class of $ω$-non-sofic groups is non-empty. Moreover, we prove that the existence of a non-sofic group implies the existence of a non-sofic group of unbounded exponent.

math.GR↗

arXiv2019

Centralizer fusion systems of central involutions in a finite group with soluble centralizer of involutions

It is a long-standing open problem raised by Starostin to describe all finite groups with soluble centralizers of involutions. One can observe that if the centralizer fusion system of an involution is nilpotent, then the centralizer of that involution is soluble. In this paper, we classify the cases when the centralizer fusion system of a central involution in a finite group whose all involutions have soluble centralizers is a nilpotent fusion system. Indeed, we analyse the case when the solvable radical has odd order and the corresponding factor group is simple.

math.GR↗