Orders of commutators and Products of conjugacy classes in finite groups
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914379031642112 |
|---|---|
| author | Tong-Viet, Hung P. |
| author_facet | Tong-Viet, Hung P. |
| contents | Let $G$ be a finite group, let $x \in G$, and let $p$ be a prime. We prove that the commutator $[x,g]$ is a $p$-element for every $g \in G$ if and only if $x$ is central modulo $\mathbf{O}_p(G)$, where $\mathbf{O}_p(G)$ denotes the largest normal $p$-subgroup of $G$. This result provides a common generalization of certain variants of both the Baer--Suzuki theorem and Glauberman's $\mathbf{Z}_p^*$-theorem. As an application, we show that if $K$ is a conjugacy class of $G$ such that $K^{-1}K = 1 \cup D \cup D^{-1}$ for some conjugacy class $D$ of $G$, then the subgroup generated by $K$ is solvable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10882 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orders of commutators and Products of conjugacy classes in finite groups Tong-Viet, Hung P. Group Theory Representation Theory 20D05, 20E45, 20D35, 20C15 Let $G$ be a finite group, let $x \in G$, and let $p$ be a prime. We prove that the commutator $[x,g]$ is a $p$-element for every $g \in G$ if and only if $x$ is central modulo $\mathbf{O}_p(G)$, where $\mathbf{O}_p(G)$ denotes the largest normal $p$-subgroup of $G$. This result provides a common generalization of certain variants of both the Baer--Suzuki theorem and Glauberman's $\mathbf{Z}_p^*$-theorem. As an application, we show that if $K$ is a conjugacy class of $G$ such that $K^{-1}K = 1 \cup D \cup D^{-1}$ for some conjugacy class $D$ of $G$, then the subgroup generated by $K$ is solvable. |
| title | Orders of commutators and Products of conjugacy classes in finite groups |
| topic | Group Theory Representation Theory 20D05, 20E45, 20D35, 20C15 |
| url | https://arxiv.org/abs/2507.10882 |