Exercise 12, § 5

Let $g$ be a Lie algebra over a field $k$ of characteristic $p > 0$. (a) Let $h$ be an integer $\leq p - 1$. $g$ is said to satisfy Engel's h-th condition if $(\mathrm{ad}\ x)^h(y) = 0$ for every ordered pair $(x, y)$ of elements of $g$. Show that this is equivalent to

for $x_1, \ldots, x_h$ in $g$ (apply the formula $(\mathrm{ad}\ x)^h = 0$ to the linear combinations of the $x_i$).

(b) Let $x, y$ be elements of $g$ such that $\Lambda_p(ax, by) = 0$ for all $a, b$ in $k$. Show that, if $\Lambda_p^{r,s}$ is the bihomogeneous component of $\Lambda_p$ of bidegree $(r, s)$, then $\Lambda_p^{r,s}(x, y) = 0$ for every ordered pair $(r, s)$. Deduce (taking $r = p - 1, s = 1$) that $(\mathrm{ad}\, x)^{p-1}(y) = 0$. In particular, if $\Lambda_p(x, y) = 0$ for $x, y$ in $g$, the Lie algebra $g$ satisfies Engel’s $(p-1)$-th condition.

(c) Let $c$ be the centre of $g$. Suppose that $(\mathrm{ad}\, x)^p = 0$ for all $x \in g$. Show that $g/c$ satisfies Engel’s $(p-1)$-th condition. (Show that $\Lambda_p(\mathrm{ad}\, x, \mathrm{ad}\, y) = 0$ for $x, y$ in $g$ and apply (b) to the Lie algebra $\mathrm{ad}\, g = g/c$.)

(d) Let $G$ be a group such that $x^p = 1$ for all $x \in G$ and let $(G_\alpha)$ be a real central filtration of $G$. The filtration $(G_\alpha)$ is restricted (Exercise 10) and $\mathrm{gr}(G)$ is a Lie $p$-algebra whose $p$-mapping is zero. Deduce that $\mathrm{gr}(G)$ satisfies Engel’s $(p-1)$-th condition.\footnote{For properties of Lie algebras satisfying an Engel condition and their applications to the “restricted Burnside problem”, see A. Kostrikin, Izv. Akad. Nauk SSSR, 23 (1959), pp. 3–34.}