Monster group

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The monster group is one of the 26 sporadic groups in group theory , a branch of mathematics . For the monster group , usually abbreviated with one of the two symbolic names and , the English names monster group , Fischer-Griess monster group or friendly giant group are often used. The unusual name of this group can be explained by the fact that it is by far the most powerful of all 26 sporadic groups .

properties

The sporadic groups are those (finitely many) finite simple groups that cannot be classified in one of the 18 (infinitely large) families of finite simple groups. There are 26 of these, and the monster group is by far the most powerful of these with a group order of

= 2 46 3 20 5 9 7 6 11 2 13 3 17 19 23 29 31 41 41 47 59 71
= 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
≈ 8 · 10 53

The order of the next smallest sporadic group, the so-called baby monster group , is about 4 · 10 33 .

The prime divisors of the order of the monster group are the supersingular prime numbers (sequence A002267 in OEIS ). The monster group is the Galois group of a polynomial with rational coefficients and can be fully characterized by specifying this polynomial.

20 of the 26 sporadic groups are subquotients ( images of subgroups ) of . According to Robert Griess, these 20 are grouped together as a happy family , and in contrast to this, the remaining 6 are referred to as parias .

Discovery story

The existence of the monster group was suspected in 1973 by Bernd Fischer and Robert Griess . In 1982 Griess succeeded in constructing the monster group as an automorphism group of a commutative , non- associative algebra on a 196883-dimensional space . 1979 formulated Simon Norton and John H. Conway , a number of assumptions about relationships between the Monster Group and the j-function ( " monstrous moonshine "), for the proof of the English mathematician Richard E. Borcherds 1998, among others on the International Congress of Mathematicians in Berlin Fields Medal .

The uniqueness of the monster was proven in 1989 by Griess, Ulrich Meierfrankenfeld and Yoav Segev.

literature

Individual evidence

  1. a b Robert L. Griess : The Friendly Giant . In: Inventiones Mathematicae . tape 69 , 1982, pp. 1-102 , doi : 10.1007 / BF01389186 ( digizeitschriften.de ).
  2. Eric W. Weisstein : Supersingular Primes . In: MathWorld (English).
  3. Eric W. Weisstein : Monster Group . In: MathWorld (English).