Walther453Fall13 Topic9 - Rhea

Monster Groups and Other Sporadic Groups

  Jill Horsfield (jbhorse@purdue.edu)
  Colin Mills (cwmills@purdue.edu)
  Andy Nelson (nelson70@purdue.edu)

1. What is a monster group?

  -Definition: A monster group is a simple, sporadic group of finite order and contains all but 6 of the other sporadic groups 
    as subgroups. Its order is: 2^46 · 3^20 · 5^9 · 7^6 · 11^2 · 13^3 · 17 · 19 · 23 · 29 · 31 · 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
  -Background: How did monster groups come about? When, who, and how did they reach this discovery?
      In 1973, Bernd Fischer and Robert Griess predicted that monster group was a simple group containing baby monster groups. 
    Robert Griess discovered the order of this monster group only a couple months after the original discovery. 
  -What makes monster groups sporadic groups?
      Sporadic groups are the finite simple groups (26) that don't fit into infinite families. 
    The largest sporadic group is the monster group.

2. Are there subgroups?

       Yes, there are subgroups.  Since the monster group is a simple group, it does not have any proper non-trivial normal subgroups.  
  -What is the subgroup structure?
       There are at least 44 conjugacy classes of maximal subgroups, as well as non-abelian simple groups of about 60 isomorphism types are either subgroups or quotients of subgroups.  The monster grip contains 20 of the 26 sporadic groups as subquotients.  The following diagram shows how they all fit together.

[1] From Symmetry and the monster by Mark Ronan.

  -What are its primes and supersingular primes?
        The supersingular primes are the set of prime numbers that divide the group order of the Monster group, namely 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71.  
  -What is a baby monster group?
  -How does existence and uniqueness relate?

3. How does the Moonshine Theory relate to Monster Groups?