Characterization of some finite simple groups by the set of orders of vanishing elements and order

• Datum: 2021
• Hauptverfasser: Askary, S., S.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2021
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/1069
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 512.5 Характеризацiя деяких скiнченних простих груп множиною порядкiв зникаючих елементiв та порядку Let $G$ be a finite group. We say that an element $g$ of $G$ is a vanishing element if there exists an irreducible complex character $X$ of $G$ such that $X(g) = 0$. Ghasemabadi, Iranmanesh, Mavadatpour (2015), present the following conjecture: Let $G$ be a finite group and $M$ a finite nonabelian simple group such that $Vo(G)=Vo(M)$ and $|G|=|M|$. Then $G \cong M $. We answer in affirmative this conjecture for $M = ^2 D_{r+1}(2)$, where $r = 2^n - 1 \geq 3$ and either $2^r+1$ or $2^{r+1}+1$ is a prime number and $M = ^2 D_{r}(3)$, where $r = 2^n + 1 \geq 5$ and either $(3^{r-1}+1)/2$ or $(3^{r}+1)/4$ is prime.