Base diagonal of a wreath product

Definition
The base diagonal of a wreath product is the subgroup comprising those members of the base power where all coordinates are equal. In the language of functions, it is the subgroup comprising the constant functions.

A subgroup is termed a base diagonal of a wreath product if it occurs as the base diagonal for some way of expressing the group as an internal wreath product.

Stronger properties

 * Weaker than::Diagonal subgroup of a direct power
 * Weaker than::Direct factor

Facts

 * If $$H$$ is the base diagonal of a wreath product in a group $$G$$, and $$H$$ is a nilpotent group of nilpotence class $$c$$, then $$H$$ is also a subnormal subgroup of subnormal depth $$c$$. In particular, if $$H$$ is abelian, then $$H$$ is normal. In fact, the following stronger statement is true: if $$H$$ has nilpotence class $$c$$, then $$[\dots [G,H],H]\dots,H]$$, with $$H$$ written $$c$$ times, is the trivial subgroup.