This level of completeness—including the class equation setup, case analysis, and referencing the cyclic quotient lemma—is what "full" means.
There is no single "official" full solution set for Chapter 4 of Abstract Algebra Dummit and Foote dummit+and+foote+solutions+chapter+4+overleaf+full
: Offers a direct PDF download of his ongoing solution project . dummit+and+foote+solutions+chapter+4+overleaf+full