arXiv subjects

Gülin Ercan

Publications and source records attributed to Gülin Ercan.

17 recordsLinked to original sources

arXiv2026

The Left-Nilpotent Residual and Its Supplements in Finite Skew Braces

Let $X$ be a finite skew brace, and let $L_\infty(X)$ be the final term of its left series. We introduce the left-nilpotent residual $\RL(X)$ and prove that $ \RL(X)=\Id_X\bigl(L_\infty(X)\bigr). $ Thus $X/\RL(X)$ is the largest left-nilpotent quotient of $X$. For skew braces of nilpotent type, we relate $\RL(X)$ to $γ_\infty(X,\cdot)$ and obtain conditions ensuring that $L_\infty(X)$ is an ideal, including the case where $|X|$ is cube-free. Our main results concern supplements to $\RL(X)$. We prove existence results using coprime action and Sylow and Hall theory, and give an example of order $18$ whose residual has no proper supplement. Finally, iterating $\RL$ yields the largest perfect subskew brace of $X$, and $X$ is KSV-solvable if and only if this subskew brace is zero.

math.GR↗

arXiv2026

Sylow theory and the nilpotency class of left nilpotent skew braces

Let $X$ be a finite left nilpotent skew brace and let $π$ be a set of primes. We show that every Hall $π$-subgroup of the multiplicative group $(X,\cdot)$ is a Hall $π$-subbrace of $X$. This extends \cite[Theorem 11]{CDDFT} by removing the solvability assumption. As an application, we obtain an upper bound for the left nilpotency class of $X$ in terms of the left nilpotency classes of its Sylow $p$-subbraces for every prime $p$ dividing the order of $X$. We also show that every $p$-subbrace of $X$ is contained in some Sylow $p$-subbrace.

math.GR↗

arXiv2026

Ideals and Solvability in Skew Braces

We investigate how nilpotency assumptions on the multiplicative group of a finite skew brace constrain its ideal structure and solvability. Our first main result shows that, if $B=(B,+,\cdot)$ is finite and $(B,\cdot)$ is nilpotent, then the additive Fitting subgroup $F(B,+)$ is a non-zero ideal of $B$. As consequences, every finite simple skew brace with nilpotent multiplicative group is isomorphic to $\Triv(C_p)$ for some prime $p$, and every such skew brace admits an ideal of prime index. In particular, \[ B*B\neq B \qquad\text{and}\qquad \partial(B) \neq B. \] We also show that nilpotency of the multiplicative group does not, in general, imply either left nilpotency or solvability. Motivated by this obstruction, we then study the solvability of two-sided skew braces, both in the finite and in the general setting. We prove that, whenever $I$ is an ideal of a two-sided skew brace $B$, the internal commutator ideal $[I,I]_I$ is again an ideal of $B$. This yields an extension theorem for solvability and implies that, for finite two-sided skew braces, solvability of the skew brace is equivalent to solvability of either the additive or the multiplicative group. In particular, every finite skew brace with abelian multiplicative group is solvable. Finally, we show that, although this equivalence fails in general for infinite two-sided skew braces, a residual form of it still survives: if the additive group is solvable, then every finite homomorphic image of the multiplicative group is solvable.

math.GR↗

arXiv2026

The Baer Transform for Skew Braces

We introduce and study the Baer transform for finite skew braces whose additive group has odd order and nilpotency class at most $2$. Starting from Baer's classical construction for nilpotent groups of odd order and nilpotency class $2$, we replace the additive group $(X,+)$ by an abelian group $(X,\oplus)$ on the same underlying set and prove that \[ Br(X)=(X,\oplus,\cdot) \] is a finite skew brace of abelian type. We show that the Baer transform preserves automorphisms, strong left ideals, ideals and central ideals, and we establish its compatibility with quotients. We then compare structural properties of $X$ and $Br(X)$. As applications, we show that direct product decompositions into ideals are preserved by the Baer transform, giving a criterion for indecomposability of skew braces. Finally, for finite $p$-skew braces with $p$ odd, we apply Thompson critical subgroups together with the Baer transform to embed suitable automorphism groups into automorphism groups of skew braces of abelian type.

math.GR↗

arXiv2026

Fixing size and Fitting height

Let $G$ be a finite solvable group on which a nilpotent group $A$ acts by automorphisms. The fixing size $\mathbf{c}(G;A)$ of $A$ on $G$ is the number of $A$-composition factors on which $A$ acts trivially in an $A$-composition series of $G$. In this paper we obtain a linear bound for the Fitting height of $G$ in terms of $\mathbf{c}(G;A)$ and $\ell(A)$ where $\ell(A)$ denotes the number of prime divisors (counted with multiplicities) of $A$, under some additional hypotheses.

math.GR↗

arXiv2024

Noncoprime action of a cyclic group

Let $A$ be a finite nilpotent group acting fixed point freely on the finite (solvable) group $G$ by automorphisms. It is conjectured that the nilpotent length of $G$ is bounded above by $\ell(A)$, the number of primes dividing the order of $A$ counted with multiplicities. In the present paper we consider the case $A$ is cyclic and obtain that the nilpotent length of $G$ is at most $2\ell(A)$ if $|G|$ is odd. More generally we prove that the nilpotent length of $G$ is at most $2\ell(A)+ \mathbf{c}(G;A)$ when $G$ is of odd order and $A$ normalizes a Sylow system of $G$ where $\mathbf{c}(G;A)$ denotes the number of trivial $A$-modules appearing in an $A$-composition series of $G$.

math.GR↗

arXiv2023

Nilpotent Residual of a Finite Group

Let $F$ be a nilpotent group acted on by a group $H$ via automorphisms and let the group $G$ admit the semidirect product $FH$ as a group of automorphisms so that $C_G(F) = 1$. We prove that the order of $γ_\infty(G)$, the rank of $γ_\infty(G)$ are bounded in terms of the orders of $γ_{\infty}(C_G(H))$ and $H$, the rank of $γ_{\infty}(C_G(H))$ and the order of $H$, respectively in cases where either $FH$ is a Frobenius group; $FH$ is a Frobenius-like group satisfying some certain conditions; or $FH=\langle α,β\rangle$ is a dihedral group generated by the involutions $α$ and $β$ with $F =\langle αβ\rangle$ and $H =\langleα\rangle$.

math.GR↗

arXiv2022

Commuting graph of a group action with few edges

Let $A$ be a group acting by automorphisms on the group $G.$ \textit{The commuting graph $Γ(G,A)$ of $A$-orbits} of this action is the simple graph with vertex set $\{x^{A} : 1\ne x \in G \}$, the set of all $A$-orbits on $G\setminus \{1\}$, where two distinct vertices $x^{A}$ and $y^{A}$ are joined by an edge if and only if there exist $x_{1}\in x^{A}$ and $y_{1}\in y^{A}$ such that $[x_{1},y_{1}]=1$. The present paper characterizes the groups $G$ for which $Γ(G,A)$ is an $\mathcal{F}$-graph, that is, a connected graph which contains at most one vertex whose degree is not less than three.

math.GR↗

arXiv2021

Good action of a nilpotent group with regular orbits

Suppose that $A$ is a finite nilpotent group of odd order acting good in the sense of \cite{EGJ} on the group $G$ of odd order. Under some additional assumptions we prove that the Fitting height of $G$ is bounded above by the sum of the numbers of primes dividing $|A|$ and $|C_G(A)|$ counted with multiplicities.

math.GR↗

arXiv2021

Extensions of several coprime results to good action case

Let $G$ and $A$ be groups where $A$ acts on $G$ by automorphisms. We say "\textit{the action of $A$ on $G$ is good}" if the equality $% H=[H,B]C_H(B)$ holds for any subgroup $B$ of $A$ and for any $B$-invariant subgroup $H$ of $G$. It is straightforward that every coprime action is a good action. In the present work we extend some results due to Ward, Gross, Shumyatsky, Jabara, and Meng and Guo under coprime action to good action.

math.GR↗

arXiv2021

Some special coprime actions and their consequence

Let a group $A$ act on the group $G$ coprimely. Suppose that the order of the fixed point subgroup $C_G(A)$ is not divisible by an arbitrary but fixed prime $p$. In the present paper we determine bounds for the $p$-length of the group $G$ in terms of the order of $A$, and investigate how some $A$-invariant $p$-subgroups are embedded in $G$ under various additional assumptions.

math.GR↗

arXiv2020

Good action on a finite group

Let $G$ and $A$ be finite groups with $A$ acting on $G$ by automorphisms. In this paper we introduce the concept of "good action"; namely we say the action of $A$ on $G$ is good, if $H=[H,B]C_H(B)$ for every subgroup $B$ of $A$ and every $B$-invariant subgroup $H$ of $G.$ This definition allows us to prove a new noncoprime Hall-Higman type theorem. If $A$ is a nilpotent group acting on the finite solvable group $G$ with $C_G(A)=1$, a long standing conjecture states that $h(G)\leq \ell(A)$ where $h(G)$ is the Fitting height of $G$ and $\ell(A)$ is the number of primes dividing the order of $A$ counted with multiplicities. As an application of our result we prove the main theorem of this paper which states that the above conjecture is true if $A$ and $G$ have odd order, the action of $A$ on $G$ is good and some other fairly general conditions are satisfied.

math.GR↗

arXiv2019

Commuting graph of $A$-orbits

Let $A$ be a finite group acting by automorphisms on the finite group $G$. We introduce the commuting graph $Γ(G,A)$ of this action and study some questions related to the structure of $G$ under certain graph theoretical conditions on $Γ(G,A)$.

math.GR↗

arXiv2019

A short note on the noncoprime regular module problem

We consider a special configuration in which a finite group $A$ acts by automorphisms on the finite group $G$, and the semidirect product $GA$ acts on the vector space $V$ by linear transformations; and discuss the existence of the regular $A$-module in $V_{_{A}}$.

math.GR↗

arXiv2019

Frobenius action on Carter subgroups

Let $G$ be a finite solvable group and $H$ be a subgroup of $Aut(G)$. Suppose that there exists an $H$-invariant Carter subgroup $F$ of $G$ such that the semidirect product $FH$ is a Frobenius group with kernel $F$. We prove that the terms of the Fitting series of $C_{G}(H)$ are obtained as the intersection of $C_{G}(H)$ with the corresponding terms of the Fitting series of $G$, and the Fitting height of $G$ may exceed the Fitting height of $C_{G}(H)$ by at most one. As a corollary it is shown that for any set of primes $π$, the terms of the $π$-series of $C_{G}(H)$ is obtained as the intersection of $C_{G}(H)$ with the corresponding terms of the $π$-series of $G$, and the $π$-length of $G$ may exceed the $π$-length of $C_{G}(H)$ by at most one. They generalize the main results of \cite{Khu}.

math.GR↗

arXiv2018

Frobenius groups of automorphisms with almost fixed point free kernel

Let $FH$ be a Frobenius group with kernel $F$ and complement $H$, acting coprimely on the finite solvable group $G$ by automorphisms. We prove that if $C_{G}(H)$ is of Fitting length $n$ then the index of the $n$-th Fitting subgroup $F_{n}(G)$ in $G$ is bounded in terms of $|C_{G}(F)|$ and $|F|.$ This generalizes a result of Khukhro and Makarenko \cite{k-m} which handles the case $n=1.$

math.GR↗

arXiv2018

On abelian group actions with TNI-centralizers

A subgroup $H$ of a group $G$ is said to be a TNI-subgroup if $N_{G}(H)\cap H^g=1$ for any $g\in G\,\backslash \,N_{G}(H).$ Let $A$ be an abelian group acting coprimely on the finite group $G$ by automorphisms in such a way that $C_G(A)=\{g\in G : g^a=g $\, for all $a\in A\}$ is a solvable TNI-subgroup of $G$. We prove that $G$ is a solvable group with Fitting length $h(G)$ is at most $h(C_G(A))+\ell(A)$. In particular $h(G)\leq \ell(A)+3$ whenever $C_G(A)$ is nonnormal. Here, $h(G)$ is the Fitting length of $G$ and $\ell(A)$ is the number of primes dividing $A$ counted with multiplicities.

math.GR↗