Mark Ronan's website
The Monster
Most finite simple groups, which are called 'symmetry atoms' in my book Symmetry and the Monster, fit into a well-understood table, a sort of periodic table. There are 26 exceptions that have no place in the table; they are called sporadic groups, and the Monster is the largest of them. It contains all but six of the others.
The size of the Monster is 246.320.59.76.112.133.17.19.23.29.31.41.47.59.71. This works out to be 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000.
The Monster is more than just the largest sporadic group. The Moonshine phenomena connect it to number theory and to string theory in mathematical physics. These connections were entirely unsuspected when the Monster first emerged, predicted as it was through one of its cross-sections, discovered by Bernd Fischer. This cross-section, later called the Baby Monster, needs more than 4,000 dimensions in which to operate, but the Monster itself needs 196,883 dimensions. This number is the product of the three largest prime numbers that are divisors of the Monster's size, namely 47, 59 and 71.
The Monster was eventually constructed by Robert Griess as the symmetry group of an algebra structure in 196,884 dimensions. His work split the space into three subspaces, and his main task was to show there were symmetries intermingling these subspaces. The dimensions of the subspaces are:
98,304 + 300 + 98,280 = 196,884
The first number 98,304 = 212 × 24
comes from the Golay code in 24 dimensions.
The second number 300 = 24 + 23 + 22 + .
. . + 3 + 2 + 1 is the dimension of the space of
24‑by‑24 symmetric matrices.
The third number 96,280 = 196,560 ÷ 2
comes from the Leech Lattice in 24 dimensions,
where there are 196,560 vertices closest to a given vertex, forming 98,280
diametrically opposite pairs.
The finite symmetry atoms are
very large, and data about each one is encoded into a character table, which is a square array of
numbers, rather like a giant sudoku puzzle. The Monster's table has 194 rows
and columns, and the Moonshine connections showed that
the first column can be used to generate an important sequence of numbers in
number theory (the coefficients of the j‑function).
Other columns can be used in a similar way, and these moonshine connections
eventually created a link to the mathematical physics of string theory.