Editing Monster group (section)

=== Computer construction ===
Martin Seysen (2022) has implemented a fast [[Python (programming language)|Python]] package named [https://mmgroup.readthedocs.io/ mmgroup], which claims to be the first implementation of the monster group where arbitrary operations can effectively be performed. The documentation states that multiplication of group elements takes less than 40 milliseconds on a typical modern PC, which is five orders of magnitude faster than estimated by [[Robert Arnott Wilson|Robert A. Wilson]] in 2013.<ref>{{cite web |url=https://mmgroup.readthedocs.io/en/latest/api.html |title=The mmgroup API reference |last=Seysen |first=Martin |access-date=31 July 2022}}</ref><ref>{{cite arXiv |last=Seysen |first=Martin |author-link= |eprint=2203.04223 |title=A fast implementation of the Monster group |class=math.GR |date=8 Mar 2022}}</ref><ref>{{cite arXiv |last=Seysen |first=Martin |author-link= |eprint=2002.10921 |title=A computer-friendly construction of the monster |class=math.GR |date=13 May 2020}}</ref><ref>{{cite arXiv |last=Wilson |first=Robert A. |author-link=Robert A. Wilson (mathematician)|eprint=1310.5016 |title=The Monster and black-box groups |class=math.GR |date=18 Oct 2013}}</ref> The mmgroup software package has been used to find two new maximal subgroups of the monster group.{{sfn|Dietrich|Lee|Popiel|2025|}}

Previously, Robert A. Wilson had found explicitly (with the aid of a computer) two invertible 196,882 by 196,882 matrices (with elements in [[GF(2)|the field of order 2]]) which together [[Generating set of a group|generate]] the monster group by matrix multiplication; this is one dimension lower than the 196,883-dimensional representation in characteristic 0. Performing calculations with these matrices was possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes.{{sfn|Borcherds|2002|p=1076}}

Wilson asserts that the best description of the monster is to say, "It is the [[automorphism group]] of the [[monster vertex algebra]]".  This is not much help however, because nobody has found a "really simple and natural construction of the monster vertex algebra".{{sfn|Borcherds|2002|p=1077}}

Wilson with collaborators found a method of performing calculations with the monster that was considerably faster, although now superseded by Seysen's abovementioned work. Let ''V'' be a 196,882 dimensional vector space over the field with 2 elements. A large subgroup ''H'' (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup ''H'' chosen is 3<sup>1+12</sup>.2.Suz.2, where Suz is the [[Suzuki group (mathematics)|Suzuki group]]. Elements of the monster are stored as words in the elements of ''H'' and an extra generator ''T''. It is reasonably quick to calculate the action of one of these words on a vector in ''V''. Using this action, it is possible to perform calculations (such as the order of an element of the monster). Wilson has exhibited vectors ''u'' and ''v'' whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an element ''g'' of the monster by finding the smallest ''i'' &gt; 0 such that ''g''<sup>''i''</sup>''u'' = ''u'' and ''g''<sup>''i''</sup>''v'' = ''v''. This and similar constructions (in different [[characteristic (algebra)|characteristics]]) were used to find some of the non-local maximal subgroups of the monster group.