A group $(G, \cdot)$ is a set with an associative binary operation, an identity $e$, and inverses. Examples: $(\mathbb Z, +)$, the symmetric group $S_n$ of permutations, $GL_n(\mathbb R)$ of invertible matrices, and the special unitary group $SU(2)$ from spin.

Lagrange's theorem: if $H \leq G$ is a subgroup of a finite group, then $|H|$ divides $|G|$. The number of cosets $[G:H] = |G|/|H|$.

A homomorphism $\varphi: G \to H$ preserves the operation: $\varphi(ab) = \varphi(a)\varphi(b)$. Its kernel $\ker \varphi$ is a normal subgroup of $G$.

First isomorphism theorem:

Quotient groups generalize "modular arithmetic" to arbitrary normal subgroups. Simple groups (no nontrivial normal subgroups) are the building blocks; their classification was a monumental 20th-century achievement.