A new criterion for a finite group G to bep-nilpotent was given

by using two characteristic subgroups D*(P) and D*e(P) for an arbitrary p-group P defined by Glauberman and Solomon in 2012.Namely

it was proved that for an odd primep

the group G was a p-nilpotent group if and only if NG(D*(P)) was a p-nilpotent group

and also

if and only if NG(D*e(P))was ap-nilpotent group too.