yes, what you describe is not a strategy that i had developed at the time! but, there is also a collection of algebraic approaches that i'm not sure i'd previously seen in algebra, & i suppose i was better prepared when analogous ideas reappeared later. e.g., for uniqueness of identity... suppose e and e' are [left & right] identities; observe that e = ee' = e'; so, they are one and the same. [similar comments for adding & subtracting a term in some analysis problems...]
still: [jumping to the next alg course] if one is proving in absalg a proposition like fermat's little theorem, then it falls straight out of lagrange's thm. for me, i'm not sure how well i cd've expected to prove f lil t prior to the prof's presentation: the idea of a coset hadn't nat'ly occurred to me, & by the time i saw the latter thm stated, it was a one line corollary.