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  "body": "\n## Statement\n\nLet $\\mathcal{S}$ be a finite thick [[generalized $n$-gon|PCT2HY9AAZ]] with $n \\ge 2$.\nThen\n$$n \\in \\set{2, 3, 4, 6, 8}.$$\n\n## Remarks\n\nWalter Feit and Graham Higman proved the theorem in 1964.\nThe proof is algebraic.\nCounting 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 $n$.\nThe multiplicities of these eigenvalues must be nonnegative integers, and this fails for all values of $n$ outside the list.\n\nEach value in the list occurs.\nFor $n = 3$ there are the [[projective planes|DANFNN8C4Y]] $PG(2, q)$, and for $n = 4$ the [[generalized quadrangles|GQJCPFSYVK]] of symplectic forms.\nFor $n = 6$ there are the split Cayley hexagons attached to the groups $G_2(q)$.\nFor $n = 8$ there are the Ree–Tits octagons attached to the groups ${}^2F_4(2^{2m+1})$.\n\nBoth hypotheses are needed.\nWithout thickness, take $n$ points and $n$ lines arranged in a cycle, the ordinary $n$-gon.\nIts incidence graph is a cycle of length $2n$, so it is a finite generalized $n$-gon for every $n$.\nWithout finiteness, there are thick generalized $n$-gons for every $n \\ge 3$, built by free constructions.\n\n## Explanations\n\n### L1 human\n\nGeneralized polygons are point-line geometries that look locally like the polygons of school geometry, but with many points on each line.\nConsider finite ones with at least three points per line and three lines per point.\nThe theorem says that the underlying polygon can only be a digon, triangle, quadrangle, hexagon or octagon.\nSo, for example, there is no finite thick generalized pentagon.\n\n### L2 human\n\nGeneralized polygons are the [[buildings|6BQJ50NEJ0]] of rank $2$, and finite buildings are tied to finite groups of Lie type.\nThe 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.\nThe method, integrality of eigenvalue multiplicities, is a standard tool of algebraic combinatorics, and it also gives conditions on the order $(s, t)$.\n",
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