Membership test-based black-box group algorithm for normality testing

Idea and outline
The idea is to first use the black-box group algorithm for small generating set-finding problem to separately compute small generating sets for $$G,H$$. Note that for $$H$$, we can enumerate all elements by first enumerating all elements of $$G$$ and then filtering the set using the membership test for $$H$$.

Once we have small generating sets for both, we can use the generating set-cum-membership test-based black-box group algorithm for normality testing. Note that the actual normality testing takes negligible time compared to the time spent computing the generating sets.