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Monster group

Monster group

A finite simple group that is an algebraic object.

The monster group is an enormous algebraic object used to explain symmetries, such as line and rotation symmetry. It is the largest of the sporadic simple group. The number of elements is 8080, 17424, 79451, 28758, 86459, 90496,17107, 57005, 75436, 80000, 00000 = 246.320. The monster group was hypothesized by Bernd Fischer and Robert Greiss. It contains all but six of the other 25 sporadic finite simple groups as subquotients, called the Happy Family.


The monster group has a connection to the j-function, modular function for SL2(Z) defined on the upper half plane. The Fourier coefficients of j-function turn out to have a decomposition in terms of dimensions of irreducible representations of monster group (196883, 21296876, ...). The monster group's first two dimensions are 1 and 196,883. The j-function's first important coefficient is 196,884, the sum of the monster group's first two dimensions. The connection between the two algebraic objects was discussed in a paper by John Conway and Simon Norton, who termed the connection 'Monstrous Moonshine'. The connection can be made with further dimensions and coefficients from the monster group and j-function.


John Conway and Simon Norton write a paper called 'Monstrous Moonshine'.

Further Resources


Group theory and why I love 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000


August 19, 2020

Mathematicians Chase Moonshine, String Theory Connections | Quanta Magazine

Erica Klarreich


March 12, 2015

What is....the Monster?

Richard E. Borcherds

October 2002


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