Exercise 22, § 1

Let $g$ be a Lie $p$-group over a ring $K$ of characteristic $p$ ($p$ prime). Let $x \mapsto x^p$ denote the $p$-mapping. For every subset $E$ of $g$, let $E^p$ denote the submodule of $g$ generated by the $x^p$, where $x \in E$. An ideal $a \subset g$ is called a $p$-ideal if $a^p \subset a$.

(a) For an ideal $a \subset g$ to be a $p$-ideal, it is necessary and sufficient that it be the kernel of a $p$-homomorphism (Exercise 20).

(b) The sum of two $p$-ideals is a $p$-ideal. The sum of a $p$-ideal and a $p$-subalgebra is a $p$-subalgebra.

(c) Let $h$ be a Lie subalgebra of $g$. The smallest Lie $p$-subalgebra containing $h$ is $h + h^p + h^{p^2} + \cdots = \overline{h}$; if we write $h_i = h + h^p + \cdots + h^{p^i}$, $h_i$ is an ideal of $\overline{h}$ and $\overline{h}/h$ is commutative. If $h$ is an ideal of $g$, so are the $h_i$ and $\overline{h}$ and the latter is the smallest $p$-ideal containing $h$.

(d) The normalizer and centralizer of an arbitrary submodule of $g$ are $p$-subalgebras of $g$.