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Julian Kaspczyk

Publications and source records attributed to Julian Kaspczyk.

6 recordsLinked to original sources
arXiv2024

The conjugacy diameters of non-abelian finite $p$-groups with cyclic maximal subgroups

Let $G$ be a group. A subset $S$ of $G$ is said to normally generate $G$ if $G$ is the normal closure of $S$ in $G.$ In this case, any element of $G$ can be written as a product of conjugates of elements of $S$ and their inverses. If $g\in G$ and $S$ is a normally generating subset of $G,$ then we write $\| g\|_{S}$ for the length of a shortest word in $\mbox{Conj}_{G}(S^{\pm 1}):=\{h^{-1}sh | h\in G, s\in S \, \mbox{or} \, s{^{-1}}\in S \}$ needed to express $g.$ For any normally generating subset $S$ of $G,$ we write $\|G\|_{S} =\mbox{sup}\{\|g\|_{S} \,|\,\, g\in G\}.$ Moreover, we write $Δ(G)$ for the supremum of all $\|G\|_{S},$ where $S$ is a finite normally generating subset of $G,$ and we call $Δ(G)$ the conjugacy diameter of $G.$ In this paper, we determine the conjugacy diameters of the semidihedral $2$-groups, the generalized quaternion groups and the modular $p$-groups. This is a natural step after the determination of the conjugacy diameters of dihedral groups, which were recently found by the first author (finite case) and by Kedra, Libman and Martin (infinite case).

math.GR↗
arXiv2023

Criteria for supersolvability of saturated fusion systems

Let $p$ be a prime number. A saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is said to be supersolvable if there is a series $1 = S_0 \le S_1 \le \dots \le S_m = S$ of subgroups of $S$ such that $S_i$ is strongly $\mathcal{F}$-closed for all $0 \le i \le m$ and such that $S_{i+1}/S_i$ is cyclic for all $0 \le i < m$. We prove some criteria that ensure that a saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is supersolvable provided that certain subgroups of $S$ are abelian and weakly $\mathcal{F}$-closed. Our results can be regarded as generalizations of purely group-theoretic results of Asaad.

math.GR↗
arXiv2022

A result on s-semipermutable subgroups of finite groups and some applications

Let $p$ be a prime number, $G$ be a $p$-solvable finite group and $P$ be a Sylow $p$-subgroup of $G$. We prove that $G$ is $p$-supersolvable if $N_G(P)$ is $p$-supersolvable and if there is a subgroup $H$ of $P$ with $P' \le H \le Φ(P)$ such that $H$ is $s$-semipermutable in $G$. As applications, we simplify the proofs of some known results and also generalize some known results.

math.GR↗
arXiv2022

On $c$-embedded subgroups of finite groups

Let $G$ be a group and $H \le K \le G$. We say that $H$ is $c$-embedded in $G$ with respect to $K$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H \cap B \le Z(K)$. Given a finite group $G$, a prime number $p$ and a Sylow $p$-subgroup $P$ of $G$, we investigate the structure of $G$ under the assumption that $N_G(P)$ is $p$-supersolvable or $p$-nilpotent and that certain cyclic subgroups of $P$ with order $p$ or $4$ are $c$-embedded in $G$ with respect to $P$. New characterizations of $p$-supersolvability and $p$-nilpotence of finite groups will be obtained.

math.GR↗
arXiv2022

On finite groups with given $ICΦ$-subgroups

A subgroup $H$ of a group $G$ is said to be an $ICΦ$-subgroup of $G$ if $H \cap [H,G] \le Φ(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $ICΦ$-subgroups of $G$. A new characterization of finite abelian groups and some new criteria for $2$-nilpotence and nilpotence of finite groups will be obtained. Moreover, we will obtain two criteria for a finite group to lie in a given solvably saturated formation containing the class of finite supersolvable groups.

math.GR↗