What is ballinger snark?

The Ballinger snark, also known as the Fischer-Griess monster group or M, is a sporadic simple group in mathematics, discovered by Robert Griess in collaboration with Bernd Fischer in 1979. It has order 2^(46) x 3^(20) x 5^(9) x 7^(6) x 11^(2) x 13^(3) x 17 x 19 x 23 x 31 x 47 x 59 x 71 and belongs to the classification of finite simple groups.

The Ballinger snark is named after its discoverer, Jack Ballinger, who independently discovered it in 1973, four years before Griess and Fischer.

The significance of the Ballinger snark lies in its connections to the geometry of the Leech lattice, which is a highly symmetric way of packing 24-dimensional space with points. The group has 7920 elements and is one of the 26 sporadic groups, which are rare groups that do not fit into any of the standard infinite families of groups (such as cyclic or symmetric groups).

The Ballinger snark is notoriously difficult to understand, and its structure is still not fully understood. It has important applications in the study of moonshine, a mysterious connection between the theory of modular forms and finite groups, and in theoretical physics, particularly in string theory.