Aug 2026· Hacettepe Journal of Mathematics and Statistics· 0 citations
Abstract
Let $\sigma=\{{\sigma_i|i\in I}\}$ be some partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. A group is said to be \textit{$\sigma$-primary} if it is a finite $\sigma_i$-group for some $i$.A subgroup $H$ of $G$ is said to be: \textit{$\sigma$-subnormal} in $G$ if there exists a subgroup chain $H=H_0\leq H_1\leq \cdots \leq H_n=G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is $\sigma$-primary for all $i=1,\ldots,n$; \textit{$\sigma$-permutably embedded} in $G$ if $H$ is $\sigma$-full and for every $\sigma_i\in \sigma(H)$, every Hall$\sigma_i$-subgroup of $H$ is also a Hall $\sigma_i$-subgroup of some $\sigma$-permutable subgroup of G.We say that a subgroup $H$ of $G$ is: \textit{$m$-$\sigma$-permutably embedded} in $G$ if$H=\langle A, B\rangle$ for some modular subgroup $A$ and $\sigma$-permutably embedded subgroup $B$ of $G$;\textit{weakly $m$-$\sigma$-permutably embedded} in $G$ if there are an $m$-$\sigma$-permutably embedded subgroup $L$ and a$\sigma$-subnormal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq L \leq H$.
Let $n\geq 1$ and let $\ell$ be an odd prime. Let $G=\mathrm{PGSp}_{2n}(\mathbb{F}_\ell)$ or $\mathrm{GSp}_{2n}(\mathbb{F}_\ell)$, and fix a faithful transitive permutation representation $\pi:G\longrightarrow S_d$. We study degree-$d$ number fields whose Galois closures have Galois group $G$ and whose associated permu...
We prove a Hall $\sigma$-subgroup analogue of the McKay conjecture for $\pi$-separable groups proposed by G. Navarro. This result simultaneously generalizes the classical McKay conjecture and its $\pi$-separable version. More precisely, if $G$ is $\pi$-separable, $p$ is a prime and $\sigma=\pi\cup\{p\}$, then it is kno...
Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write $\mathcal{O}_S$ for the ring of $S$-integers of $K$ and $\mathcal{O}_S^\times$ for its unit group. Let $\pi : X \rightarrow \mathbb{P}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\operatorname{Red}(\pi)$ the se...
Let $G$ be a countable locally finite abelian group, and let \[ G_1\leq G_2\leq\cdots, \qquad \bigcup_{i\geq1}G_i=G, \] be any filtration of $G$ by finite subgroups. For $A\subseteq G$, let $\mathcal P(A)$ denote the set of all finite subset sums of elements of $A$, and let $2G:=\{2g:g\in G\}$. We prove that $|2G|=\inf...
For a finite group $H$, let $\nu(H)$ denote the maximum order of a nilpotent subgroup of $H$. We prove that every finite solvable transitive permutation group $P$ on a finite set $\Omega$ has a subset $\Delta\subseteq\Omega$ such that $|P:P_\Delta|\ge\nu(P_\Delta)$. We also prove that if a finite solvable group $H$ act...
Given an ideal $\mathcal I$ on $\omega$, a subgroup $H$ of the unit circle $\mathbb T$ is said to be $\mathcal I$-characterized if there exists a sequence of integers $(a_n:n\in\omega)$ such that $$ H= \left\{ x\in\mathbb T: \mathcal I\text{-}\lim_{n\to\infty}a_nx=0 \right\}. $$ We investigate the descriptive complexit...
P. Leonetti· 0 citations
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