The influence of the nilpotentlizers on group structur

For a finite group G and an element x∈ G, the subset nilG(x)=∈ G <x,y>~~ is ~~ nilpotent\ 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 nilG(x) generates a maximal subgroup of G and the simple commutator of weight $2 ~~or ~~3$ of elements of nilG(x) is equal to $1$ or |nilG(x)|= pⁿ, where p is prime and $n=1, 2$. Then G is a solvable group.