4-Regular prime graphs of nonsolvable groups

素图正则的有限群

Finite groups whose prime graphs are regular

设G是有限群，Irr（G）是G的所有不可约复特征标的集合.设cd（G）是G的所有特征标度的集合，ρ（G）表示能整除G的某些特征标度的素数的集合.与G相关联的素图Δ（G）是顶点集为ρ（G）且两个不同素数p和q之间存在边的图当且仅当乘积pq整除G的某个特征度数。本文证明了有限群G的素图Δ（G）是3-正则的当且仅当它是四顶点完全图.

Regular orbits of symmetric and alternating groups

Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2. In this paper, we classify the pairs $(G,V)$ for which $G$ has a regular orbit on $V$ where $G$ is a covering group of a symmetric or alternating group and $V$ is a faithful irreducible $FG$-module such that the order of $F$ is prime and divides the order of $G$.

Regular orbits of sporadic simple groups

Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2. Let $G$ be a covering group of an almost simple group whose socle $T$ is sporadic, and let $V$ be a faithful irreducible $FG$-module where $F$ has prime order dividing $|G|$. We classify the pairs $(G,V)$ for which $G$ has no regular orbit on $V$, and determine the minimal base size of $G$ in its action on $V$. To obtain this classification, for each…

Steinberg-like characters for finite simple groups

Let $G$ be a finite group and, for a prime $p$, let $S$ be a Sylow $p$-subgroup of $G$. A character $χ$ of $G$ is called $\Syl_p$-regular if the restriction of $χ$ to $S$ is the character of the regular representation of $S$. If, in addition, $χ$ vanishes at all elements of order divisible by $p$, $χ$ is said to be Steinberg-like. For every finite simple group $G$ we determine all primes $p$ for which $G$ admits a Steinberg-like character, except for alternating groups in characteristic~2. Moreover, we determine all primes for which $G$ has a projective $FG$-module of dimension $|S|$, where $F…