isomorphism theorems
When you collapse part of an algebraic object by a homomorphism, the part you kill (the kernel) and the part you keep (the image) are tightly linked. The isomorphism theorems are the precise rules that tell you, again and again, that “the source modulo what got killed equals the image.” They are the basic grammar of quotients.
There are three, stated here for groups (analogues hold for modules, rings, and more). First: if f : G -> H is a homomorphism, then G / ker(f) is isomorphic to im(f). Second (the diamond): for a subgroup A and a normal subgroup N of G, AN/N is isomorphic to A/(A ∩ N). Third (the correspondence/quotient theorem): for normal subgroups N ⊆ M of G, (G/N)/(M/N) is isomorphic to G/M, and there is an inclusion-preserving bijection between subgroups of G containing N and subgroups of G/N.
These theorems let you compute with quotients without ever building them by hand: to identify G/N, find a homomorphism out of G with kernel N and read off the image. They are not deep so much as foundational — almost every later structural argument, from the Jordan-Hölder theorem to module theory, is an organized application of them. The honest caveat is bookkeeping: the second and third theorems require the relevant subgroups to be normal where stated, and forgetting that is the usual source of errors.
The determinant det : GL(n, R) -> R^* is a homomorphism with kernel SL(n, R), so by the first isomorphism theorem GL(n, R) / SL(n, R) is isomorphic to R^*. Likewise the map Z -> Z/nZ has kernel nZ, giving Z/nZ as Z modulo nZ — the prototypical quotient.
First isomorphism theorem applied to the determinant.