Sporadic group

A sporadic group is a finite simple group outside the infinite families of cyclic groups of prime order, alternating groups, and groups of Lie type. There are 26 such groups.

Finite group: A group with finitely many elements. Sporadic groups are finite groups singled out by their failure to fit the standard infinite families.
Monster group: The largest sporadic simple group, with order about 8 × 10^53. Its extraordinary size makes it the most famous endpoint of the sporadic list.
Émile Mathieu: A French mathematician who discovered five sporadic simple groups in the 1860s. His permutation groups supplied the first known exceptions to the infinite families.
Leech lattice: A 24-dimensional even unimodular lattice with no vectors of squared length two. Several sporadic groups arise from symmetries of this exceptionally symmetric lattice.
Classification proof project: The extensive body of arguments establishing the classification of finite simple groups. Its length and complexity make verification and exposition a continuing mathematical undertaking.
Simple group: A nontrivial group whose only normal subgroups are the identity subgroup and itself. Simplicity is the defining structural condition shared by every sporadic group.
Mathieu group: One of five sporadic simple groups introduced by Émile Mathieu in the nineteenth century. The Mathieu groups are the earliest discovered sporadic examples.
Zvonimir Janko: A Croatian mathematician whose 1960s work initiated the modern discovery of sporadic simple groups. His proposed new simple group renewed the search for exceptional finite examples.
Golay code: An error-correcting code with parameters that give unusually high information density and correction ability. The Mathieu groups act naturally on the coordinates of the binary Golay code.
Monster group representations: Linear and module actions of the Monster group on vector spaces and related algebraic structures. Representations help explain how the largest sporadic group acts in moonshine and other settings.

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Monstrous moonshineNarrower topic: The Monster is the largest and most intricate member of this exceptional collection.