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For a finite group G and an element x G, the subset nilG (x) =\y G ~~ 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.
Ahmadkhah et al. (Sat,) studied this question.
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