Base of a wreath product with diagonal action

Definition
A subgroup $$G$$ of a group $$L$$ is termed a base of a wreath product with diagonal action if $$L$$ can be expressed as an internal wreath product with diagonal action with $$G$$ as base. In other words, $$L$$ is an internal semidirect product of a direct power $$H$$ of $$G$$ (with $$G$$ as one of the factors) and a subgroup $$K_1 \times K_2$$ where $$K_1$$ acts by coordinate permutations and $$K_2$$ acts diagonally by automorphisms on each coordinate.

Stronger properties

 * Weaker than::Direct factor
 * Weaker than::Base of a wreath product
 * Weaker than::Complemented normal subgroup

Weaker properties

 * Stronger than::Direct factor of normal subgroup
 * Stronger than::Direct factor of complemented normal subgroup
 * Stronger than::2-subnormal subgroup

Facts

 * A normal subgroup of a characteristic subgroup of the base of a wreath product with diagonal action is still a 2-subnormal subgroup.


 * Weaker than::Base of a wreath product