On $AAM$'s Conjecture for $D_{n}(3)$

Liangcai Zhang ; Wenmin Nie ; Dapeng Yu

Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 4

Résumé

The noncommuting graph of a finite nonabelian group $G$, denoted $\nabla(G)$, is defined as follows: its vertices are the non-central elements of $G$, and two vertices are adjacent when they do not commute. Problem $16.1$ in \emph{the Kourovka Notebook } contains the following conjecture: If $M$ is a finite nonabelian simple group and $G$ is a group such that $\nabla(G)\cong \nabla (M)$, then $G\cong M$. % (see \cite{KhM}). The validity of this conjecture is still unknown for most of finite simple groups with connected prime graphs even though it is known to hold for all finite simple groups with disconnected prime graphs and only a few of finite simple groups with connected prime graphs, for example, $A_{10}$ and $L_{4}(9)$. % and so on (see \cite{WS,ZS3}). In the present paper, it is proved that the finite simple group of Lie type $D_{n}(3)$, where $n\geq5$ is an odd integer or $n=p+1$ for a prime $p>3$, is quasirecognizable by its prime graph. In particular, $AAM$'s conjecture is true for it. Thus it is an example of an infinite series of finite simple groups recognizable by their noncommuting graphs, whose prime graphs are connected for some $n$. %Method used in this paper can be applied to some other finite simple groups.

Détail

Classification : 20D05, 20D06, 20D60