Symmetry and Exceptional Structure: From the Polygon to the Monster

The dihedral group of order 2n collects the n rotations and n reflections of a regular polygon with n sides, so you can check its rules by turning and flipping a paper polygon. It is the standard first example of a non-abelian group, where the order of combining two symmetries matters, and it reappears in crystallography and in frieze and wallpaper patterns.

The symmetric group on n elements holds every permutation of a set and has exactly n factorial elements. Cayley's theorem shows that every group is isomorphic to a subgroup of some symmetric group, which makes this the universal setting for finite symmetry, and it is abelian only when n is 2 or smaller.

The general linear group of degree n is the set of invertible n by n matrices under matrix multiplication, and for degree two or higher it is not abelian. Évariste Galois constructed the group over a prime field and computed its order in 1832, and its subgroup of matrices with determinant one is the special linear group.

The braid group on n strands is the group of equivalence classes of braids under ambient isotopy, introduced explicitly by Emil Artin in 1925. Alexander's theorem lets every knot be seen as the closure of a braid, and Artin's presentation of the group matches the Yang-Baxter equation in mathematical physics.

The Fano plane is the smallest finite projective plane, with seven points and seven lines, three points on every line and three lines through every point. It is named for Gino Fano, who wrote on the foundations of geometry in 1892, it can be built from the two-element field, and it is denoted PG(2, 2).

The octonions are an eight-dimensional normed division algebra, built from the quaternions by the Cayley-Dickson construction. They were discovered in December 1843 by John T. Graves, inspired by Hamilton's quaternions, and unlike the quaternions they are nonassociative, yet they connect to the exceptional Lie groups.

The E8 lattice is the unique positive-definite, even, unimodular lattice of rank eight, and its 240 shortest nonzero vectors form the root system of the E8 Lie group. H. J. S. Smith showed its existence in 1867, and Maryna Viazovska proved in 2016 that spheres centered on its points form the densest packing in eight dimensions.

The Leech lattice is an even unimodular lattice in twenty-four dimensions, introduced by John Leech in 1965 while studying sphere packings, with each sphere touching 196,560 neighbors. Its automorphism group is the Conway group, and through the Griess algebra construction its symmetry connects to the Monster group.

The Monster is the largest of the 26 sporadic simple groups, with an order of about 8.08 times ten to the 53rd power. Fischer and Griess predicted it independently in the early 1970s and Griess constructed it between 1980 and 1982, and it anchors monstrous moonshine, its link to the j-invariant that Borcherds proved in 1992.