[bwtsjfnu] The complexity of number theory and group theory

Number theory owes its richness to a delicate and complicated interaction between addition and multiplication.  To pick a famous example, 1 + 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 641 * 6700417.  Where did those random-looking numbers on the right "come from"?  (Euler.)

Group theory owes its richness the same way, giving us, for example, the incredibly complicated classification of finite simple groups.  Despite the group axioms only using "addition", we end up defining a multiplication-like operation between groups (not elements) via the concepts of subgroups and group quotients.