arXiv subjects

M. Yasir Kızmaz

Publications and source records attributed to M. Yasir Kızmaz.

7 recordsLinked to original sources

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

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↗

arXiv2023

A generalization of Alperin fusion theorem and its applications

Let $\mathcal F$ be a saturated fusion system on a finite $p$-group $S$, and let $P$ be a strongly $\mathcal F$-closed subgroup of $S$. We define the concept ``$\mathcal F$-essential subgroups with respect to $P$" which are some proper subgroups of $P$ satisfying some technical conditions, and show that an $\mathcal F$-isomorphism between subgroups of $P$ can be factorised by some automorphisms of $P$ and $\mathcal F$-essential subgroups with respect to $P$. When $P$ is taken to be equal $S$, Alperin-Goldschmidt fusion theorem can be obtained as a special case. We also show that $P\unlhd \mathcal F$ if and only if there is no $\mathcal F$-essential subgroup with respect to $P$. The following definition is made: a $p$-group $P$ is strongly resistant in saturated fusion systems if $P\unlhd \mathcal F$ whenever there is an over $p$-group $S$ and a saturated fusion system $\mathcal F$ on $S$ such that $P$ is strongly $\mathcal F$-closed. It is shown that several classes of $p$-groups are strongly resistant, which appears as our third main theorem. We also give a new necessary and sufficient criteria for a strongly $\mathcal F$-closed subgroup to be normal in $\mathcal F$. These results are obtained as a consequences of developing a theory of quasi and semi-saturated fusion systems, which seems to be interesting for its own right.

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

On the influence of the fixed points of an automorphism to the structure of a group

Let $α$ be a coprime automorphism of a group $G$ of prime order and let $P$ be an $α$-invariant Sylow $p$-subgroup of $G$. Assume that $p\notin π(C_G(α))$. Firstly, we prove that $G$ is $p$-nilpotent if and only if $C_{N_G(P)}(α)$ centralizes $P$. In the case that $G$ is $Sz(2^r)$ and $PSL(2,2^r)$-free where $r=|α|$, we show that $G$ is $p$-closed if and only if $C_G(α)$ normalizes $P$. As a consequences of these two results, we obtain that $G\cong P\times H$ for a group $H$ if and only if $C_G(α)$ centralizes $P$. We also prove a generalization of the Frobenius $p$-nilpotency theorem for groups admitting a group of automorphisms of coprime order.

math.GR↗

arXiv2019

A Generalization of Hall-Wielandt Theorem

Let $G$ be a finite group and $P\in Syl_p(G)$. We denote the $k$'th term of the upper central series of $G$ by $Z_k(G)$ and the norm of $G$ by $Z^*(G)$. In this article, we prove that if for every tame intersection $P\cap Q$ such that $Z_{p-1}(P)<P\cap Q<P$, the group $N_G(P\cap Q)$ is $p$-nilpotent then $N_G(P)$ controls $p$-transfer in $G$. For $p=2$, we sharpen our results by proving if for every tame intersection $P\cap Q$ such that $Z^*(P)<P\cap Q<P$, the group $N_G(P\cap Q)$ is $p$-nilpotent then $N_G(P)$ controls $p$-transfer in $G$. We also obtain several corollaries which give sufficient conditions for $N_G(P)$ to controls $p$-transfer in $G$ as a generalization of some well known theorems, including Hall-Wielandt theorem and Frobenius normal complement theorem.

math.GR↗

arXiv2018

Finite Groups Having Nonnormal T.I. Subgroups

In the present paper, the structure of a finite group $G$ having a nonnormal T.I. subgroup $H$ which is also a Hall $π$-subgroup is studied. As a generalization of a result due to Gow, we prove that $H$ is a Frobenius complement whenever $G$ is $π$-separable. This is achieved by obtaining the fact that Hall T.I. subgroups are conjugate in a finite group. We also prove two theorems about normal complements one of which generalizes a classical result of Frobenius.

math.GR↗