The discussion centers on the connection between the kissing number in 24 dimensions (196,560) and the Monster group, which is the largest of the 26 sporadic groups with approximately 8×10^53 elements. The Monster group serves as the symmetry group for a 196,883-dimensional object, while the value 196,884 appears in the Fourier expansion of the J-invariant. The Leech lattice plays a crucial role in determining the sphere packing in 24 dimensions, and John Conway's work in the late 1960s established significant links between these mathematical concepts, culminating in the "monstrous moonshine" conjecture, later proven by Richard Borcherds.
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No. The largest of the 26 sporadic groups, but there are bigger regular ones, ##\mathbb{Z}_p## for example with an arbitrary large prime.WWGD said:I never understood the big deal. Is the monster group just the largest simple group?
fresh_42 said:Which group are you talking about? The monster group has a several magnitude larger order.
The connection is the Leech lattice. It determines the locations of the spheres for the 24-dimensional packing. Conway investigated the Leech lattice as he considered the groups.Labyrinth said:The Monster group is the symmetry group of a 196,883 dimensional object. The value 196,884 appears as a coefficient in the Fourier expansion of the J-invariant.
These numbers are quite close to the kissing number in 24 dimensions: 196,560. I've heard that there is a connection but I can't quite find where this was all worked out, assuming it was.
https://de.wikipedia.org/wiki/John_Horton_Conway#Wirken said:Conway discovered three new sporadic finite simple groups in the late 1960s, the Conway groups named after him, when he was studying the Leech lattice. He also simplified the construction of the last and largest sporadic group found, the "monster" (but preferred by the discoverer to be called "friendly giant"). In a famous work with his doctoral student Simon Norton from the late 1970s he pointed to the connections between the (dimensions of the irreducible) representations of the monster and the expansion coefficients of the elliptical module function, called "monstrous moonshine" after the title of their essay (she followed an observation by John McKay). Many of the suspected connections were later proven by Conway's PhD student Richard Borcherds, who received the Fields Medal for it. With his research group in Cambridge, Conway published the Atlas of Finite Groups.