# What are normal closure of a complement of a subgroup and intersection of all normal subgroups?

What are normal closure of a complement of a subgroup and intersection of all normal subgroups?

If X is a nonempty subset of a group G, define the normal closure X complement of X to be the intersection of all normal subgroups of G that contain X; that is, core H = {g in G | g in aHa^-1 for all a in G} = intersections of {aHa^-1 | a in G}

Sorry first time posting here and I'm not good with codes.

If $$H$$ is a proper subgroup of $$G$$, then the subgroup generated by its complement, $$\langle G-H \rangle=G$$ (this follows from the fact that a group cannot be the union of two proper subgroups). Hence the normal closure, $$(G-H)^G=\langle G-H \rangle^G=G$$.