Mathematical One-liners

This is a collection of succinct but wonderfully satisfying theorems. The punchlines are hidden; try to guess the blanks! Toggle theorem names. Toggle spoilers.

• Any bounded complex differentiable function is constant.
  Liouville’s theorem

• Any matrix is similar to an upper triangular matrix.
  Jordan normal form

• The integral of a holomorphic function around a loop vanishes identically.
  Cauchy’s integral theorem

• Strict subsets of which are nonempty, open, and simply connected are related by a conformal mapping.
  Riemann Mapping Theorem

• A complete ordered field is isomorphic to the real line.

• Every field is a subfield of the surreal numbers.

• All groups are permutation groups.
  Cayley’s theorem

• The order of a finite group is divisible by the orders of its subgroups.
  Lagrange’s theorem

• A finite Abelian group is a direct sum of prime-order cyclic groups.
  Fundamental theorem of finite Abelian groups

• The -modules are exactly the Abelian groups.

• A square matrix satisfies its own characteristic polynomial.
  Cayley–Hamilton theorem

• There are arbitrarily long arithmetic progressions of primes.
  Green–Tao theorem

• Every proper arithmetic sequence contains infinitely many primes.
  Dirichlet’s Theorem

• Two random infinite graphs are isomorphic with probability one.

• Every normed division algebra is isomorphic to , , or .
  Hurwitz’s Theorem