Theorems with an extraordinary exception or a small number of sporadic exceptions

Question

The Whitney graph isomorphism theorem gives an example of an extraordinary exception: a very general statement holds except for one very specific case.

Another example is the classification theorem for finite simple groups: a very general statement holds except for very few (26) sporadic cases.

I am looking for more of this kind of theorems-with-not-so-many-sporadic-exceptions

(added:) where the exceptions don't come in a row and/or in the beginning - but are scattered truly sporadically.

(A late thanks to Asaf!)

Answer 1

Every automorphism of $S_n$ is inner if $n \neq 6$.

P.S. That $S_6$ has an `essentially' unique outer automorphism is quite a non-obvious fact.

Answer 2

I've posted this answer to other questions, but it's worth repeating:

The topological manifold $\mathbb{R}^n$ has a unique smooth structure up to diffeomorphism as long as $n \neq 4$. However, $\mathbb{R}^4$ admits uncountably many exotic smooth structures.

Edit: Other thoughts:

• The h-cobordism theorem holds for $n= 0, 1, 2$ and $n \geq 5$. It is open for $n = 3$ and false (smoothly) for $n=4$.

• I've heard that many theorems in number theory only work for field characteristic $\neq 2$ (but my number theory background is sadly lacking).

• $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic iff $n = 1, 2, 4, p^k, 2p^k$, where $p$ is an odd prime, $k \geq 1$. (Granted, there aren't a finite number of exceptions here, but I wanted to mention it anyway.)

Answer 3

Carmichael's Theorem: Every Fibonacci number $F_n$ has a prime factor which does not divide any earlier Fibonacci number, except for $F_1 = F_2 = 1$, $F_6 = 8$ and $F_{12} = 144$.

This is a special case of his more general result: Let $P,Q$ be nonzero integers such that $P^2 > 4Q$, and consider the Lucas sequence $D_1 = 1$; $D_2 = P$; $D_{n+2} = P \cdot D_{n+1} - Q \cdot D_n$. Then all but finitely many $D_n$ have a prime factor which does not divide $D_m$ for any $m < n$; the only possible exceptions are $D_1$, $D_2$, $D_6$ and $D_{12}$.

Answer 4

Every simply connected subregion of $\mathbb C$ is conformally equivalent to the unit disk, with the exception of $\mathbb C$ itself.

Answer 5

For squarefree positive integer $d$, every imaginary quadratic field $\mathbb{Q}(\sqrt{−d})$ has class number greater than 1 unless $d$ is equal to one of the Heegner numbers: 1, 2, 3, 7, 11, 19, 43, 67, 163.

Answer 6

Here's a neat result that holds for all finite groups except if the Monster shows up as a quotient:

Let $G$ be a finite group and $p,q$ primes dividing $|G|$. If $G$ contains no element of order $pq$, then either

1. the Sylow $p$-subgroups or the Sylow $q$-subgroups are abelian, or
2. $G/O_{\{p,q\}'}(G)$ is the Monster and $\{p,q\}=\{5,13\}$ or $\{7,13\}$.

[source]

Answer 7

The only consecutive positive integer powers are 8 and 9.

This was conjectured by Eugène Catalan in 1844 and proved by Preda Mihăilescu in 2002.

Answer 8

Map Color Theorem: For any surface $\Sigma$ with Euler characteristic $c \leq 0$, with the exception of the Klein bottle, $$\chi(\Sigma) = \frac{1}{2} (7 + \sqrt{49 - 24c}),$$ where $\chi$ is the chromatic number.

Answer 9

Mitchell's theorem: A primitive complex reflection group is either the symmetric group $S_n \subseteq GL_{n-1}(\mathbb{C})$ or one of $34$ exceptions.

Definitions: a complex reflection group is a finite subgroup $W$ of $GL_n(\mathbb{C})$ for some $n$, that is generated by reflections. A reflection is an invertible matrix with codimension one fixed space. A complex reflection group $W \subseteq GL_n(\mathbb{C})$ acting on $V=\mathbb{C}^n$ is imprimitive if there is a decomposition $V=V_1 \oplus V_2$ such that for each $w \in W$, and $i=1,2$, $w(V_i) \subseteq V_j$ for some $j$. Otherwise it is primitive.

Answer 10

How about the Big Picard theorem? http://en.wikipedia.org/wiki/Picard_theorem

If a function $f:\mathbb{C}\to \mathbb{C}$ is analytic and has an essential singularity at $z_0\in \mathbb{C}$, then in any open set containing $z_0$, $f(z)$ takes on all possible complex values, with at most one possible exception, infinitely often.

Answer 11

There are several theorems in topology and geometry that hold for manifolds of all dimensions save 3 and sometimes 4. For example, of Euclidean space except $\mathbb{R}^3$ and sometimes $\mathbb{R}^4$. There are also some conjectures proved for $\mathbb{R}^n$, n ≠ 3 (resp {3,4}), but which are open questions for n = 3 ({3,4}). Wikipedia describes it thus:

[T]he cases $N = 3$ or $4$ have the richest and most difficult geometry and topology. There are, for example, geometric statements whose truth or falsity is known for all N except one or both of 3 and 4. N = 3 was the last case of the Poincaré conjecture to be proved.

Using the regular polytopes as an example:

• There are exactly 3 regular (convex) polytopes, the n-simplexes, -cubes, and -orthoplexes which exist in all dimensions, except 3 and 4, which have several additional Platonic solids and regular polychora without higher analogues (ignoring the trivial cases of dimensions 1 and 2).
• There are no regular non-convex polytopes except in dimensions 2, 3, and 4.

Further,

• Moise's theorem holds only in 3 dimensions.
• There is exactly 1 differentiable structure on $\mathbb{R}^n$, up to diffeomorphism, except for n = 4, which has infinitely many. See "Bizarre Fact 48".

In a sense, 3- and 4-dimensional spaces are privileged. This has implications in the philosophy of physics: why did space (apparently) have 3 spatial dimensions rather than 2 or 527?

See also 3-manifold, special phenomena of 4-manifolds, low-dimensional topology.

Answer 12

Vinogradov's theorem that all but a finite number of odd numbers are the sum of three primes.

Not sure about the current state of Goldbach's conjecture.

A theorem at about my level of math: All primes greater than two are odd.

Answer 13

Not sure if this fits the scope of the question, it's hardly as advanced as all the other suggestions but I find it quite elegant and surprising:

There exists no positive integer sandwiched between a perfect square and a perfect cube, with the sole exception of $26$ - equivalent to saying that the only solution of $a^2 \pm 2 = b^3$ is $(5, 3)$.

Of course this is just one among many Diophantine equations which admit only one solution, but it has a special significance when viewed from a less abstract point of view.

Another nice one:

For any prime $p$, the product of its primitive roots is congruent to $1$ modulo $p$, except for $p = 3$ for which the product is equal to $2$ (result due to Gauss).

Answer 14

Polynomial equations of every degree have no general solution in radicals, with the exception of degrees 1, 2, 3 and 4.

Answer 15

The only spheres which admit almost complex structures are $S^2$ and $S^6$.

Answer 16

Brownian motion is transient in every dimension, with the exception of dimensions 1 and 2.

Answer 17

The "sausage catastrophe" for finite sphere packings comes to mind:

For 1..55 spheres a linear "sausage" is the optimal packing, for higher numbers some cluster packing is optimal.

Answer 18

How about the Ax-Kochen theorem?

Every homogeneous polynomial of degree $d$ in $n$ variables with $n>d^2$ has a non-trivial zero in $\mathbb{Q}_p$ for all but finitely many $p$... and the finite set of exceptions depends on the degree $d$.

Here are some things we know about the finite exception set for various $d$: http://en.wikipedia.org/wiki/Ax-Kochen_theorem#Exceptional_primes

This is a special case of the more general fact that any first-order sentence (in the language of valued fields) which is true of all but finitely many Laurent series fields $\mathbb{F}_p((t))$ is true of all but finitely many $p$-adic fields $\mathbb{Q}_p$. Model theory gives us many more examples of the principle "true in char $0$" = "true in char $p$ for large enough $p$".

Answer 19

Fermat's Last Theorem: For any positive integer $n$, except $n = 1, 2$, there is no solution of positive integers $(x, y, z)$ to the equation $$x^n + y^n = z^n.$$

Answer 20

Here's a beautiful theorem of Peter J Cameron from the theory of designs:

Theorem. If symmetric $2-(v,k,\lambda)$ design $\mathscr{D}$ extends, then it is one of the following :

1. $2-(4\lambda+3,\;\; 2\lambda+1,\;\; \lambda )$¹
2. $2-((\lambda+2)(\lambda^2+4\lambda+2), \;\;\lambda^2+3\lambda+1, \;\;\lambda)$
3. $2-(495,39,3)$

This appeared in the 1973 paper of Prof. P J Cameron [Cam]. When it was stated, the existence of the design of parameters $2-(111, 11, 1)$² was yet undecided. It has been now proved with an extensive computer search $[10]$ that this design does not exist.

Some (perhaps) Useful References.

[Cam] Cameron P. J., Extending Symmetric Designs, Journal of Combinatorial Theory, Series A Vol. 14, Issue 2 (Mar., 1973), pp. 215-220.

$[10]$ Lam C. W. H., Thiel L. H., Swiercz S., The Non-existence of Finite Projective Plane of Order 10 Can. J. Math., XLI (1989), pp. 1117-1123.

_{¹ Note that these are the parameters of a Hadamard $2$-design.
²Some readers will recognise that this is a projective plane of order 10.}

Answer 21

Any number of theorems about quadratic forms are true over fields where the characteristic is not equal to 2. My favorite reference for the subject, where this sense of "nothing works the same in characteristic 2" is made clear as early as the "Notes to the Reader," is Lam's Introduction to the Theory of Quadratic Forms over Fields. For an even more elementary example of this phenomenon, note that the quadratic formula only holds (or even makes sense) for fields of characteristic not equal to 2.

Answer 22

The classification of simple finite dimensional Jordan algebras has exactly one exceptional case.

That of simple Lie algebras has five exceptions.

(All this over sensible fields, of course)

There are 3 regular polyhedra in every dimension, except in dimesions 2, 3 and 4.

Answer 23

$A_n$ has no nontrivial normal subgroup unless $n=4$.

Answer 24

No free group is amenable, with the exception of $\mathbb Z$.

Answer 25

Every family of graphs that is closed under minors has at least one forbidden minor, with the exception of the family of all graphs.

Answer 26

Other than the sphere, all closed surfaces have list chromatic number equal to chromatic number.

Answer 27

Every prime number is odd, with the exception of 2.

Answer 28

There exists an Aperiodic Tiling of $\mathbb{R}^d$ for all dimensions excepting $d=1$. For $d=1$ it is easy to show that an aperiodic set of tiles cannot exist.

Also, for $d \geq 3$ there exists an aperiodic single tile, the problem is still open in $d=2$ (A non-connected single aperiodic tile was discovered 2 years ago, but usually we ask for connected tiles).

(Edit: $2$-dimensional case was solved: https://en.wikipedia.org/wiki/Einstein_problem

Answer 29

A graph is planar, unless it contains a copy of $K_5$ or $K_{3,3}$.

Every subgroup of $\mathbb{Q}$ is residually finite, except $\mathbb{Q}$ itself.

The only two perfect squares in the sequence $\lbrace\displaystyle\sum_{i=1}^n i^2\rbrace_n$ are $1^2$ and $70^2$.

Answer 30

The sphere $S^n$ is simply connected if and only if $n\geq 2$.

Answer 31

The modular curves $X_0(N)$ have no exceptional automorphism except for $N=37$, $63$ and $108$.

Recall the modular group $$\Gamma_0(N):=\left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \ : \ c \equiv 0 \pmod n \right\} $$ which acts in the upper half-plane $$\mathbb{H}:=\{z\in \mathbb{C}\ : \ \operatorname{Im}(z)>0\}$$ via Moebius transformations. The quotient $\Gamma_0(N)\backslash \mathbb{H}$ is analytically isomorphic to an affine algebraic curve over $\mathbb{C}$, with associated non-singular projective curve $X_0(N)$, the classical modular curve. Suppose the genus of this curve is $\ge 2$ (which happens for all $N$ except a finite number of them, including all $N>36$, $N\ne 49$).

The normalizer $N(n)$ of $\Gamma_0(N)$ inside $\operatorname{SL}_2(\mathbb{Z})$ acts naturally as automorphism in $X_0(N)$ and it is natural to ask if they are all automorphism. Ogg proved in 1974 that this is the case if $N$ is square-free and $N\ne 37$, which it has an exceptional automorphism not coming from $N(37)$. In 1988, Kenku and Momose proved that there was non exceptional automorphism except for $N=37$ and may be for $N=63$. Elkies in 1990 showed that there really was an exceptional automorphism for $N=63$.

But in 2011, after more than 20 years, Michael Colin Harrison, when revising the proof by Kenku and Momose found out that he could not discard the case $N=108$ by their arguments. After doing some computations with the help of MAGMA he was able to find an exceptional automorphism for $N=108$, later confirmed by Elkies. The result was only published in the arxiv.

For more information you can consult the preprint by Harris: A new automorphism of $X_0(108)$.

Answer 32

All triangular graphs $T_n$ are determined by their spectrum except for $n=8$.

Answer 33

The number of partitions of $n\geq 0$ into parts congruent to $\pm 1, \pm 4, \pm 5, \pm 6, \pm 7, \pm 9, \pm 11, \pm 13, \pm 16, \pm 21, \pm 23, \pm 28$ (mod 66) is equal to the number of partitions of $n\geq 0$ into parts congruent to $\pm 1, \pm 4, \pm 5, \pm 6, \pm 7, \pm 9, \pm 11, \pm 14, \pm 16, \pm 17, \pm 27, \pm 29$ (mod 66) except for $n=13$. See here and here for this and many similar results.

Answer 34

This paper http://www.jstor.org/stable/2323537?origin=crossref&seq=1#page_scan_tab_contents shows that in some sense, $\mathbb R^n$ can only have a cross product defined on it if $n=3$ or $7$.

Answer 35

Just about every single theorem in arithmetic that concerns prime numbers fails for $2$.

Answer 36

If $\sin x$ is rational then $x/\pi$ is transcendental, except when $\sin x$ is $0$, $\pm\frac12$, or $\pm1$.

Answer 37

Given $R>0$ and $a\in \mathbb{C}$. For every $n\in \mathbb{Z}$ except $n=-1$ we have:

\begin{align} \oint_{|z-a|=R} (z-a)^n \mathrm d z=0 \end{align}

Answer 38

Let $X=(V,E)$ be a finite graph with $N:=|V|$ vertices and adjacency matrix d.

Say that a square matrix $u$ with entries in a unital $\mathrm{C}^*$-algebra is a magic unitary if its entries are projections which sum to the identity on each row and column: $$\sum_{k=1}^N u_{ik}=\sum_{k=1}^Nu_{kj}=1.$$

Let $C(G^+(X))$ be the universal $\mathrm{C}^*$-algebra generated by the entries of a magic unitary $u\in M_N(C(G^+(X)))$ subject to the relation $du=ud$.

Consider the cyclic graph $C_N$. Then:

For $N\neq 4$, $C(G^+(C_N))$ is commutative and finite dimensional; but for $N= 4$, $C(G^+(C_N))$ is non-commutative and infinite dimensional.

The significance of this is contained in the quip:

The only cyclic graph with non-trivial quantum automorphisms is $C_4$.

To explain all this, including the choice of notation, please see the book, Quantum Permutation Groups, by Teo Banica.

The original reference is:

Teo Banica, Quantum automorphism groups of small metric spaces, Pacific J. Math. 219 (2005), >27-51.

Answer 39

A triangle group is a group generated by three elements $a$, $b$, $c$ of finite orders $p$, $q$, $r$ such that the product $a\cdot b\cdot c$ is identity.

Such a group can have arbitrarily large (countable!) order unless $1/p+1/q+1/r>1$. (These are not exactly finitely many cases since on has $(p,q,r)=(2,2,n)$ as a possibility; however, there are only finitely many $(p,q,r)$ that solve this inequality.)

Answer 40

$\prod\limits_{k=0}^{n}\binom{2\color{red}{n}}{k}$ is divisible by $\prod\limits_{k=0}^n\binom{2\color{red}{k}}{k}$ only if $n=1,2,5$. Proof

For example, for $n=5$, the product of the blue numbers is divisible by the product of the orange numbers.

Answer 41

Have a look at the Strong Law of Small Numbers:

When two numbers look equal, it ain't necessarily so.