seminars:alge
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| * **October 7**\\ <html> <span style="color:blue;font-size:120%"> No Algebra Seminar </span></html> \\ | * **October 7**\\ <html> <span style="color:blue;font-size:120%"> No Algebra Seminar </span></html> \\ | ||
| - | * **October 14**\\ <html> <span style="color:blue;font-size:120%"> Hung Tong-Viet (Binghamton University) </span></html> \\ **//Title//** \\ \\ <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP> | + | * **October 14**\\ <html> <span style="color:blue;font-size:120%"> Hung Tong-Viet (Binghamton University) </span></html> \\ **//Orders of commutators and Products of conjugacy classes in finite groups//** \\ \\ <WRAP center box 90%> **//Abstract//**: Let $G$ be a finite group, $x\in G$, and let $p$ be a prime. In this talk, we explore conditions that forces $x$ to lie in certain characteristic subgroups of $G$. In particular, we prove that the commutator $[x,g]$ is a $p$-element for all $g\in G$ if and only if $x$ is central modulo $O_p(G)$, the largest normal $p$-subgroup of $G$. This result unifies and generalizes aspects of both the Baer-Suzuki theorem and Glauberman's $Z_p^*$-theorem. Additionally, we show that if $x\in G$ is a $p$-element and there exists an integer $m\ge 1$ such that for every $g\in G$, the commutator $[x,g]$ is either trivial or has order $m$, then the subgroup generated by the conjugacy class of $x$ is solvable. As an application, we confirm a conjecture of Beltran, Felipe, and Melchor: if $K$ is a conjugacy class in $G$ such that the product $K^{-1}K=1\cup D\cup D^{-1}$ for some conjugacy class $D$, then the subgroup generated by $K$ is solvable. </WRAP> |
| * **October 21**\\ <html> <span style="color:blue;font-size:120%"> Inna Sysoeva (Binghamton University) </span></html> \\ **//Title//** \\ \\ <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP> | * **October 21**\\ <html> <span style="color:blue;font-size:120%"> Inna Sysoeva (Binghamton University) </span></html> \\ **//Title//** \\ \\ <WRAP center box 90%> **//Abstract//**: Text of Abstract </WRAP> | ||
seminars/alge.1759257311.txt · Last modified: 2025/09/30 14:35 by tongviet