Groups nearly of prime exponent and nearly Engel Lie algebras (Q1265454)

Let \(L\) be a finite Lie algebra having prime characteristic \(p\) (which can be considered as a finite-dimensional Lie algebra over the prime field \(GF(p)\)). \(L\) is said to satisfy the Engel condition if \([\dots[[x,y],y],\dots,y]= x\text{ ad}(y)^n= 0\) for every \(x,y\in L\). The authors prove that if \(S\) is a vector subspace of \(L\) and if the probability \(\alpha\) that two elements in \(S\) satisfy the Engel identity \(E_n\) is greater than \((2^{n+1}- 1)/2^{n+1}\) then the Engel identity \(E_n\) is satisfied in \(S\). They also prove that if \(G\) is a finite \(p\)-group and that the probability that an element of \(G\) has order dividing \(p\) is \(>(3^p- 2)/(3^p- 1)\), then \(L(G)\), the Lie algebra associated to \(G\), satisfies the Engel condition \(E_{p-1}\).