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Rulin Shen

Publications and source records attributed to Rulin Shen.

7 recordsLinked to original sources
arXiv2026

On Zappa's question in the case of alternating groups

In 1962, Guido Zappa asked whether a non-trivial coset of a Sylow $p$-subgroup of a finite group could contain only elements whose orders are powers of $p$. Marston Conder gives a positive answer to this question in the case of $p=5$. It is known that the smallest group satisfying the conditions of this problem must be a non-abelian simple group. In this paper, we prove that the smallest group of the Zappa problem could not be an alternating simple group for any prime $p$.

math.GR↗
arXiv2025

Characterization of $\PSL(2,q)$ by the number of singular elements

Given a finite group $G$, let $\pi(G)$ denote the set of all primes that divide the order of $G$. For a prime $r \in \pi(G)$, we define $r$-singular elements as those elements of $G$ whose order is divisible by $r$. Denote by $S_r(G)$ the number of $r$-singluar elements of $G$. We denote the proportion $S_r(G)/|G|$ of $r$-singular elements in $G$ by ${\mu_r}(G)$. Let $\mu(G) := {\{\mu_r}(G) | r\in \pi(G)\}$ be the set of all proportions of $r$-singular elements for each prime $r$ in $\pi(G)$. In this paper, we prove that if a finite group $G$ has the same set $\mu(G)$ as the simple group $\PSL(2,q)$, then $G$ is isomorphic to $\PSL(2,q)$.

math.GR↗
arXiv2024

On small Sylow numbers of finite groups

Let $G$ be a finite group and $n_p(G)$ the number of Sylow $p$-subgroups of $G$. In this paper, we prove if $n_p(G)<p^2$ then almost all numbers $n_p(G)$ are a power of a prime.

math.GR↗
arXiv2023

On Thompson Problem

In 1987, the second author of this paper reported his conjecture, all finite simple groups $S$ can be characterized uniformly using the order of $S$ and the set of element orders in $S$, to Prof. J. G. Thompson. In their communications, Thompson posed his problem about the judgment of solvability of finite groups $G$. In this paper we give a positive answer for Thompson's problem if the prime graph of $G$ is not connection.

math.GR↗
arXiv2022

Finite groups in which every maximal subgroup is nilpotent or normal or has $p'$-order

Let $G$ be a finite group and $p$ a fixed prime divisor of $|G|$. Combining the nilpotence, the normality and the order of groups together, we prove that if every maximal subgroup of $G$ is nilpotent or normal or has $p'$-order, then (1) $G$ is solvable; (2) $G$ has a Sylow tower; (3) There exists at most one prime divisor $q$ of $|G|$ such that $G$ is neither $q$-nilpotent nor $q$-closed, where $q\neq p$.

math.GR↗
arXiv2020

The second minimum/maximum value of the number of cyclic subgroups of finite $p$-groups

Let $C(G)$ be the poset of cyclic subgroups of a finite group $G$ and let $\mathcal{P}$ be the class of $p$-groups of order $p^n$ ($n\geq 3$). Consider the function $α:\mathcal{P}\longrightarrow (0, 1]$ given by $α(G)=\frac{|C(G)|}{|G|}$. In this paper, we determine the second minimum value of $α$, as well as the corresponding minimum points. Further, since the problem of finding the second maximum value of $α$ was completely solved for $p=2$, we focus on the case of odd primes and we outline a result in this regard.

math.GR↗