The nilpotency class of finite groups of exponent 𝑝

We investigate the properties of Lie algebras of characteristic p p which satisfy the Engel identity x y n = 0 x{y^n} = 0 for some n > p n > p . We establish a criterion which (when satisfied) implies that if a a and b b are elements of an Engel- n n Lie algebra L L then a b n − 2 a{b^{n - 2}} generates a nilpotent ideal of L L . We show that this criterion is satisfied for n = 6 , p = 7 n = 6,\,p = 7 , and we deduce that if G G is a finite m m -generator group of exponent 7 7 then G G is nilpotent of class at most 51 m 8 51{m^8} .