Walther453Fall13 Topic9 - Rhea

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

By definition, a monster group is a simple, sporadic group of finite order that 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}  $

Upon thinking about a monster group, one might ask what a sporadic group or even more generally what a group is?
A group is formed when a set of elements together can combine any two elements to form a third element (also in the set) while satisfying associativity,
identity, and invertibility. One of the most familiar examples of a group is the set of integers with the addition operation. In this case, the addition of
any two integers forms another integer. Notation would look like:

    $   \left\vert \mathrm{G} \right\vert = \left( \mathbb{Z}, + \right)  $

This seems simple when a group contains one, two or even three elements, but as groups grow they become more and more complex.
To reduce complexity, mathematicians break groups into smaller groups and study subgroups, simple groups, and finite groups like sporadic groups.
In group theory, sporadic groups have no normal subgroups except the subgroup containing only of the identity element itself.

So what makes monster groups sporadic groups? Sporadic groups are the finite simple groups that don't fit into infinite families.
There are 26 sporadic simple groups. The monster group is the largest group of this type.

So how did monster groups come about?
In 1973, Bernd Fischer and Robert Griess began studying sporadic groups and predicted that the
monster group was a simple sporadic group containing baby monster groups as subgroups. They thought its formation was much like other groups of
larger size. Robert Griess discovered the order of this monster group only a couple months after the original discovery.

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?
The baby Monster group is a group of order:

    $  \left\vert \mathrm{B} \right\vert = 2^{41} \times 3^{13} \times 5^6 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 23 \times 31 \times 47  $

    $  = 4,154,781,481,226,426,191,177,580,544,000,000  $

    $ \approx 4 \times 10^{33}  $

Like the Monster Group, the Baby Monster Group group is one of the sporadic simple groups. In fact, it is the sporadic simple group with the second highest order, below only the Monster Group.

The double cover of the Baby Monster is the centralizer of an element of order 2 in the Monster group. The centralizer of a subset, B, of a group, M, is the set of elements of M that commute with each element of B.
The Baby Monster Group has 30 classes of maximal subgroups

-How does existence and uniqueness relate?
The Monster Group was predicted by Bernd Fischer and Robert Griess in 1973 as a simple group containing a double cover of Fischer's Baby Monster Group as a centralizer of an involution. Shortly after, Griess found the order of the Monster Group M by using the Thompson order formula. Griess then went on to construct M as the automorphism group of the Griess algebra, which is a 196,884-dimensional commutative non associative algebra. This construction was later simplified by John Conway and Jacques Tits. Therefore the Monster Group's existence was proven.

A combined effort from Griess, Meierfrankenfeld, and Segev was able to give the first complete published proof on the uniqueness of the Monster Group, in which they showed that a group with the same centralizers of involutions as the Monster Group is isomorphic to the Monster Group.

Before the above proof was published, Norton announced that he was able to prove the uniqueness of the Monster Group from the existence of a 196,883-dimensional faithful representation, although Norton never published the details of his proof.

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 the coefficient of $ 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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