Feit–Higman theorem

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 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 $PG(2, q)$, and for $n = 4$ the generalized quadrangles 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.

Going deeper

For a student

Generalized polygons are the buildings 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 $(s, t)$.