Diagonal subgroup of a wreath product with a cyclic permutation group is divisible by all primes dividing the order of the cyclic group

Statement
Let $$L$$ be the external wreath product of a group $$G$$ by a finite cyclic group of order $$n$$ acting in the natural way as permutations on a set of size $$n$$. Let $$K$$ be the diagonal subgroup corresponding to this wreath product in $$L$$, so $$K \cong L$$. Then, the following are true:


 * 1) For any $$k \in K$$, there exists $$l \in L$$ such that $$l^n = k$$.
 * 2) Let $$\pi$$ be the set of all primes dividing $$n$$. Then, $$K$$ is divisible by $$\pi$$ in $$L$$.

Applications

 * Every group is a subgroup of a divisible group
 * Every pi-group is a subgroup of a divisible pi-group

Proof sketch
Suppose $$t$$ is a generator for the acting cyclic group. Any element $$k \in K$$ can be written as a diagonal tuple $$(g,g,\dots,g)$$ for $$g \in G$$. Suppose $$h = (g,1,1,\dots,1)$$. Let $$l = ht$$. Then, $$l^n= k$$.