The influence of the nilpotentlizers on group structur

For a finite group $G$ and an element $x\in G$, the subset $$ni{l}_G(x)=\{y\in G\mid <x,y>isnilpotent\}$$ is called nilpotentizer of $x$ in $G$. In this paper, we give two solvabilty criteria for a finite group by the structure and the size of nilpotentizer of an element on finite group. In fact, we show that if there exists an element $x$ of $G$ such that $ni{l}_G(x)$ generates a maximal subgroup of $G$ and the simple commutator of weight $2or3$ of elements of $ni{l}_G(x)$ is equal to $1$ or $\mathrm{\mid }ni{l}_G(x)\mathrm{\mid }=p^n$, where $p$ is prime and $n=1,2$. Then $G$ is a solvable group.