This is a collection of succinct but wonderfully satisfying theorems. Leave a comment if you have a good one!
The punchlines are hidden; try to guess the blanks!
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Liouville’s theorem
Any bounded complex differentiable function is constant.Jordan normal form
Any matrix is similar to an upper triangular matrix.Cauchy’s integral theorem
The integral of a holomorphic function around a loop vanishes identically.Riemann Mapping Theorem
Strict subsets of $ℂ$ which are nonempty, open, and simply connected are related by a conformal mapping.A complete ordered field is isomorphic to the real line.
Every field is a subfield of the surreal numbers.
Cayley’s theorem
All groups are permutation groups.Lagrange’s theorem
The order of a finite group is divisible by the orders of its subgroups.Fundamental theorem of finite Abelian groups
A finite Abelian group is a direct sum of prime-order cyclic groups.The $ℤ$-modules are exactly the Abelian groups.
Cayley–Hamilton theorem
A square matrix satisfies its own characteristic polynomial.Green–Tao theorem
There are arbitrarily long arithmetic progressions of primes.Dirichlet’s Theorem
Every proper arithmetic sequence contains infinitely many primes.Two random infinite graphs are isomorphic with probability one.
Hurwitz’s Theorem
Every normed division algebra is isomorphic to $ℝ$, $ℂ$, $ℍ$ or $𝕆$.