Walther453Fall13 Topic9 - Rhea

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?

Moonshine

   (n.) Foolish talk or ideas. 

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 (as well as 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. Since then all of the connections between modular
forms and the monster group, along with most of the other finite simple sporadic groups, have collectively become known
as "moonshine".
Still, though, you may be wondering why they decided on the term "moonshine". Well let's take a deeper look at
the history of the monstrous moonshine, including its outline, and then clear that up at the end.

The story begins with these three equalities:

   $  1 = 1  $
   $  196884 = 196883 + 1  $
   $  21493760 = 21296876 + 196883 + 1  $

The numbers on the right side of the equalities are associated with the Monster group itself.
The numbers on the left side of the equalities come from the modular function $ j(z) $, which is a function on the upper half plane:

   z ∈ H = {z ∈ ℂ: Im(z)> 0}

$ j(z) $ is the quintessential example of a modular function that appears in complex analysis and number theory, and that
transforms "nicely" such that:

   $  j(z)=j(z+1)=j(\frac{-1}{z})  $

If we let $ q=e^{2\pi iz} $ we see that j(z) can be expressed as a q-series with integer coefficients.
There does not exist a "good" definition of a q-series, so it will suffice to say it is a series with q's in the summands,
and j(z) now looks like this:

   $ j(z)= \frac{1}{q}+744+196884q+21493760q^{2}+.... $

Now for the name. In the late 1970's, mathematician John McKay told John Conway that this coefficient $ q $ was
precisely the dimension of the Griess algebra, which is 196884. The Griess algebra is a commutative non-associative algebra on
a real vector space that has the Monster group as its automorphism group. In response to this claim by McKay, Conway
exclaimed that this was "moonshine!" And the name stuck. In sum, the term refers to both the Monster group (M) itself
as well as the perceived absurdity of the relationship between M and modular functions!

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