Abelian and MWNSCDIN implies SCDIN

Statement

If $H$ is an abelian MWNSCDIN-subgroup of a group $G$, then $H$ is a SCDIN-subgroup of $G$.

Proof

Proof idea

The idea is that since $H$ is abelian, any subset of size one in $H$ is a normal subset. Thus, given any subset $A$ of $H$, we can view $A$ as the collection of its singleton subsets, and use the fact that $H$ is MWNSCDIN to conclude that it is SCDIN.

Proof details

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