---
tag: P2V0XS4F63
kind: statement
role: theorem
name: Feit–Higman theorem
msc: ["51E12", "51E24", "05B25"]
uses: [PCT2HY9AAZ, DANFNN8C4Y, GQJCPFSYVK, 0BQAA2BK3Q, 6BQJ50NEJ0]
about: [PCT2HY9AAZ]
sources:
  - url: https://en.wikipedia.org/wiki/Generalized_polygon
---

## Statement

Let $\mathcal{S}$ be a finite thick [[generalized $n$-gon|PCT2HY9AAZ]] with $n \ge 2$.
Then
$$n \in \set{2, 3, 4, 6, 8}.$$

## Remarks

Walter Feit and Graham Higman proved the theorem in 1964.
The proof is algebraic.
Counting paths in the incidence graph gives a matrix identity for the point-line incidence matrix, from which its [[eigenvalues|0BQAA2BK3Q]] can be computed in terms of the order $(s, t)$ and $n$.
The multiplicities of these eigenvalues must be nonnegative integers, and this fails for all values of $n$ outside the list.

Each value in the list occurs.
For $n = 3$ there are the [[projective planes|DANFNN8C4Y]] $PG(2, q)$, and for $n = 4$ the [[generalized quadrangles|GQJCPFSYVK]] of symplectic forms.
For $n = 6$ there are the split Cayley hexagons attached to the groups $G_2(q)$.
For $n = 8$ there are the Ree–Tits octagons attached to the groups ${}^2F_4(2^{2m+1})$.

Both hypotheses are needed.
Without thickness, take $n$ points and $n$ lines arranged in a cycle, the ordinary $n$-gon.
Its incidence graph is a cycle of length $2n$, so it is a finite generalized $n$-gon for every $n$.
Without finiteness, there are thick generalized $n$-gons for every $n \ge 3$, built by free constructions.

## Explanations

### L1 human

Generalized polygons are point-line geometries that look locally like the polygons of school geometry, but with many points on each line.
Consider finite ones with at least three points per line and three lines per point.
The theorem says that the underlying polygon can only be a digon, triangle, quadrangle, hexagon or octagon.
So, for example, there is no finite thick generalized pentagon.

### L2 human

Generalized polygons are the [[buildings|6BQJ50NEJ0]] of rank $2$, and finite buildings are tied to finite groups of Lie type.
The theorem explains why the labels of the Coxeter diagram of a finite thick building lie in $\set{2, 3, 4, 6, 8}$: its rank $2$ residues are finite thick generalized polygons.
The method, integrality of eigenvalue multiplicities, is a standard tool of algebraic combinatorics, and it also gives conditions on the order $(s, t)$.