arXiv subjects

Changguo Shao

Publications and source records attributed to Changguo Shao.

4 recordsLinked to original sources

arXiv2025

Supersolvable subgroups of order divisible by 3

We determine the structure of the finite non-solvable groups of order divisible by $3$ all whose maximal subgroups of order divisible by $3$ are supersolvable. Precisely, we demonstrate that if $G$ is a finite non-solvable group satisfying the above condition on maximal subgroups, then either $G$ is a $3'$-group or $G/{\bf O}_{3'}(G)$ is isomorphic to ${\rm PSL}_2(2^p)$ for an odd prime $p$, where ${\bf O}_{3'}(G)$ denotes the largest normal $3'$-subgroup of $G$. Furthermore, in the latter case, ${\bf O}_{3'}(G)$ is nilpotent and ${\bf O}_2(G)\leq {\bf Z}(G)$.

math.GR↗

arXiv2025

Indices of non-supersolvable maximal subgroups in finite groups

Two classic results, due to K. Doerk and P. Hall respectively, establish the solvability of those finite groups all of whose maximal subgroups are supersolvable, and the solvability of finite groups in which all maximal subgroups have prime or squared prime index. In this note we describe the structure of the non-solvable finite groups whose maximal subgroups are either supersolvable or have prime or squared prime index.

math.GR↗

arXiv2025

Extensions of a theorem of P. Hall on indexes of maximal subgroups

We extend a classical theorem of P. Hall that claims that if the index of every maximal subgroup of a finite group $G$ is a prime or the square of a prime, then $G$ is solvable. Precisely, we prove that if one allows, in addition, the possibility that every maximal subgroup of $G$ is nilpotent instead of having prime or squared-prime index, then $G$ continues to be solvable. Likewise, we obtain the solvability of $G$ when we assume that every proper non-maximal subgroup of $G$ lies in some subgroup of index prime or squared prime.

math.GR↗

arXiv2024

p-Nilpotent maximal subgroups in finite groups

Let $p$ be a prime number and suppose that every maximal subgroup of a finite group is either $p$-nilpotent or has prime index. Such group need not be $p$-solvable, and we study its structure by proving that only one nonabelian simple group of order divisible by $p$, which belongs to the family ${\rm PSL}_n(q)$, can be involved in it. For $p=2$, we specify more, and in fact, such simple group must be isomorphic to ${\rm PSL}_2({r^a})$ for certain values of the prime $r$ and the parameter $a$.

math.GR↗