

Groups and Group Convolutions - misiti3780
http://colah.github.io/posts/2014-12-Groups-Convolution/

======
pash
Coincidentally, I'm reading Roy McWeeny's Classic book _Symmetry: An
Introduction to Group Theory and Its Applications_ [0] right now. It's a
"leisurely but reasonably thorough" (as its author describes it) look at
finite groups with an eye towards scientific applications, a good practical
introduction for readers with a decent math background.

Hermann Weyl's book _Symmetry_ [1] is a more truly leisurely introduction,
much more a discussion of the ideas than a math book.

0\.
[http://www.amazon.com/gp/product/0486421821/](http://www.amazon.com/gp/product/0486421821/)

1\. [http://www.amazon.com/Symmetry-Hermann-
Weyl/dp/0691023743](http://www.amazon.com/Symmetry-Hermann-Weyl/dp/0691023743)

------
natejenkins
Although I'm a physicist group theory was one of my favorite undergraduate
courses. At the risk of sounding like an idiot to a mathematician, I will
attempt to explain one of the cooler parts of an extremely cool subject. There
are finite and infinite groups. An example of the former is the modulo3
addition operator over the elements [0,1,2]. The identity element is 0, 0+0=0,
0+1=1, 0+2=2. The inverse of 0 is 0, of 1 is 2 (1 + 2 mod 3 = 0) and of 2 is 1
(2 + 1 mod 3 = 0). Addition is also associative so we have that covered. An
example of an infinite group would be addition over the integers.

Anyway, focusing on finite groups, mathematicians spent many years trying to
classify finite groups into different types. For instance, every finite group
of prime length is a cyclic group
([http://groupprops.subwiki.org/wiki/Group_of_prime_order](http://groupprops.subwiki.org/wiki/Group_of_prime_order)).
This is already very cool since it implies that for each prime number, there
is exactly one group with that number of elements, anything that looks like
another group can be mapped 1:1 with the cyclic group.

There are many other types of finite groups and the vast majority (this is an
understatement) can be classified as one type or another. At some point
mathematicians realized that although there are an infinite number of finite
groups, maybe they could classify all finite groups into one type or another
and set about doing so, with the caveat that there is a finite number of
exceptions to this classification, and these exceptions are called sporadic
groups
([http://en.wikipedia.org/wiki/Sporadic_group](http://en.wikipedia.org/wiki/Sporadic_group)).

Here is where my knowledge is the fuzziest but also where things get awesome.
Using a form of magic only available to mathematicians, it was predicted that
an extremely large sporadic group existed somewhere out there, avoiding
classification into one of the standard group types. I am guessing that
certain hints to its existence were popping up here and there but have no idea
as to what those hints might look like. In physics, particles have been
predicted before (and at least one time using group theory:
[http://en.wikipedia.org/wiki/Eightfold_Way_%28physics%29](http://en.wikipedia.org/wiki/Eightfold_Way_%28physics%29)),
these predictions usually stem from attempting to explain experimental
results. Mathematics lacks these experimental results so these sorts of
predictions are all the more impressive.

Anyway, back to this extremely large group. It was given the name the Monster
Group and an upper bound was given as to its size. If I recall correctly (I
cannot find a citation), this upper bound was considered the largest useful
number of all time (any child can name a number larger than a number given to
them, but using it in some novel way is not so easy). Eventually the Monster
Group was found, fulfilling the prediction, and its size given as
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000, far
below the upper bound but still quite large nonetheless
([http://en.wikipedia.org/wiki/Monster_group](http://en.wikipedia.org/wiki/Monster_group)).

Also, mathematicians succeeded in classifying finite groups
([http://en.wikipedia.org/wiki/Classification_of_finite_simple...](http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups)),
it only took several hundred articles and tens of thousands of pages to do so.

~~~
felixcq
To anyone interested in reading more about the how the classification of
finite simple groups was "discovered" over time (the main families, then each
new sporadic group, and finally proving that all of them have indeed been
found), I strongly recommend reading the following book: Symmetry and the
Monster, by Mark Ronan.

It's not so much about the math than the history behind the quest, with took
about 50 years to be completed.

