Diagonal subgroup of a wreath product

Definition

A subgroup $G$ of a group $L$ is termed a diagonal subgroup of a wreath product if we can express $L$ as an internal wreath product (the internal version of external wreath product) with $G$ as the diagonal subgroup corresponding to the base direct power.

Relation with other properties

Related properties

• Base of a wreath product: Note that for any wreath product, the base and diagonal are isomorphic as abstract groups. However, they need not be automorphic subgroups. In fact, while the base is always a 2-subnormal subgroup, the diagonal need not be subnormal at all, and in fact it is subnormal if and only if (as an abstract group) it is a nilpotent group (note that the whole group still need not be nilpotent).