Difference between revisions of "Walther453Fall13 Topic9" - Rhea

Revision as of 12:39, 29 November 2013

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

-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:

    $  \left\vert \mathrm{M} \right\vert = 2^{46} \times 3^{20} \times 5^9 \times 7^6 \times 11^2 \times 13^3 \times 17 \times 19 \times 23 \times 29 \times 31 \times 41 \times 47 \times 59 \times 71  $

    $  = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000  $

    $ \approx 8 \times 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.

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?

The moonshine theory, also referred to as "monstrous moonshine", is a term coined by the brilliant mathematicians

John Conway and Simon P. Norton back in 1979.  They used this to describe the unexpected connection between the

monster group M, along with other sporadic finite groups, and the seemingly-unrelated modular functions.  While much

of their research was purely conjectural at the time, many of these conjectures have since been proven.  Their research
 formed the basis upon which the knowledge of this subject has grown, and since then all of the connections between modular
 forms and the monster group, along with most of the other finite simple sporatic groups, have collectively become known

as "moonshine".

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