Recognition of some finite simple groups by the orders of vanishing elements
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Let $G$ be a finite group. A vanishing element of $G$ is an element $g\in G$ such that $\chi(g) = 0$ for some irreducible complex character $\chi$ of $G$. Denote by ${\rm Vo}(G)$ the set of the orders of vanishing elements of $G$. A finite group all of whose elements have prime power order is called a CP-group. Generally, a finite group $G$ is called a VCP group if every element in ${\rm Vo}(G)$ is a prime power. Here, we classify completely the non-solvable VCP-groups and show that, except for $A_7$, the non-solvable VCP-groups coincide with the non-solvable CP-groups. Moreover, as a consequence, we give a new characterization of the simple VCP-groups, namely, if $G$ is a finite group and $M$ is a finite non-abelian simple VCP-group except for L$_2$(9) such that ${\rm Vo}(G) = {\rm Vo}(M)$, then $G\cong M$. In addition, we also prove that if ${\rm Vo}(G) = {\rm Vo}(L_2(9))$, then either $G\cong L_2(9)$, or $G \cong NA$, where $A\cong SL_2(4)$ and $N$ is an elementary abelian 2-group and a direct sum of natural $SL_2(4)$-modules.
DANDAN LIU^{1} JINSHAN ZHANG^{1} ^{}
Volume 131, 2021
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