MWNSCDIN-subgroup

Symbol-free definition
A subgroup of a group is termed a MWNSCDIN-subgroup if it is a defining ingredient::multiple weak normal subset-conjugacy-determined subgroup inside its defining ingredient::normalizer relative to the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a MWNSCDIN-subgroup if, given a collection of normal subsets $$A_i, i \in I$$ and $$B_i, i \in I$$ of $$H$$, and an element $$g \in G$$ such that $$gA_ig^{-1} = B_i$$ for all $$i \in I$$, there exists $$x \in N_G(H)$$ such that $$xA_ix^{-1} = B_i$$ for all $$i \in I$$.

Stronger properties

 * Weaker than::Pronormal subgroup:

Weaker properties

 * Stronger than::WNSCDIN-subgroup
 * Stronger than::Subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed