The idea that a subgroup can generate the original group is an interesting idea. The definition of a subgroup doesn’t mention anything about infinite subgroups, so they should be a possibility. (Later they are explicitly mentioned but only with cyclic subgroups.)
The center of a group is the set . Every example Hungerford gives only has the center being the identity element e or it fails to have a center. Theorem 7.5 part one states that the set G has a unique identity element. The identity element obviously meets this characteristic, but what other cases are valid?