Group Theory: Unveiling The Monster

In April 2020, the mathematics community lost one of it’s legends to the unforgiving coronavirus – John Conway. Conway was known for his contributions in numerous areas of mathematics including number theory, game theory, topology, probability, algebra, and most notably, his invention of the Game of Life.

In a heart-warming interview with Numberphile, he mentioned:

“The one thing I’d like to know before I die is why the Monster Group exists.”

You’re probably wondering, what is the Monster Group? And how are we supposed to figure this out if the genius himself hasn’t?

Here’s what we know so far.

‘Monster’s Inc Ride and Go Seek’ by Joel via Flickr under Creative Commons Liscence 2.0.

It is monstrous.

Discovered by Robert Griess in 1973 and later named by Conway, the Monster Group is larger than any group you possibly could imagine – your local sports club, the population of the Earth, and even the number of stars in the universe. So how many ‘members’ exactly does it have? The answer is:

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

What monstrosity of a group could possibly be this big? And how could a number so arbitrarily large be so finite?

 

It is a group of symmetries.

Group theory revolves around symmetries. In a simple example, I will present you this ordinary equilateral triangle and close my eyes.

‘Equilateral triangle’ by me (Canis Nugroho).

You have to find as many unique ways to flip or rotate the triangle without me noticing.

Let’s start with rotations. You can rotate the triangle 120 degrees, 240 degrees, and do nothing. Yes, doing nothing is a valid action. You could also rotate it 360 degrees, but the corners end up in the same position as doing nothing.

I’ll illustrate this and label the corners to make it easier to visualize.

‘Rotational Symmetries of Equilateral Triangle’ by me.

You can also flip the triangle around the three different axes.

‘Reflectional Symmetries of Equilateral Triangle’ by me.

Together, these six actions form the group of symmetries of our triangle.

We can find the symmetries for all kinds of objects: polygons, snowflakes, and the list goes on. But the Monster Group is based on symmetries of a very abstract object, one with an astounding 196,883 dimensions. No more, no less.

 

It is a Simple Group.

Some symmetries can be also be produced by combining multiple others. Going back to our triangle, you can rotate the triangle 240 degrees, then flip it on the a-axis. This would be the same as flipping it on the c-axis.

‘Combinations of Symmetries’ by me.

Combining symmetries is fundamental in stripping groups into their simplest forms – the Simple Group. A Simple Group is defined if:

  • It contains our favorite ‘do nothing’ element.
  • All elements are products of combinations of other symmetries in the group.

For our triangle, one Simple Group would be the rotations – 120 degrees, 240 degrees, and doing nothing. Doing any combination of these rotations would simply lead to another rotation of triangle – there’s no way to flip it.

The Monster Group is no exception. Every single one of its symmetries obeys the Simple Group properties and none of them are redundant.

 

It has children.

Rather than focusing on specific groups, mathematicians classify groups into families based on similar properties. One example of a family is the rotational symmetries for all polygons with a prime number of sides.

However, there are 26 lone groups known as the Sporadic Groups. The Monster is the largest of these.

Nineteen of these groups, including one called the Baby Monster Group, are considered to be children of the Monster. Griess described these 20 groups as the ‘Happy Family’.

 

It is difficult to work with.

More than 40 years after it’s discovery, there is still so much of the Monster Group that remains a mystery, even to the great mathematicians such as Conway. Most of this is because it is difficult to visualise and probe the structure of this mysterious object.

“I haven’t got 196,883-dimensional eyes,” Conway explained. “So I’ll never see it.”

 

It has some connections to other fields of mathematics.

The entire concept of the Monster Group seems arbitrary, as nobody even knows why it’s there. But Conway argues that “[The Monster Group] has too many intriguing properties for it to be a coincidence.”

In 1978, John McKay found a coefficient one bigger than 196,883 in his completely unrelated field of math called modular forms and elliptical functions. Whilst most mathematicians let this slide, Conway refused to believe that it happened by chance, calling it “Moonshine.

More recently, the Monster has been found to have some connection with String Theory.

Even in his final years, John Conway had faith in the Monster Group the way teachers do in students. The Monster was never completely unveiled in his lifetime, but if we keep pursuing it, we could in ours.

 

 

Further Reading and Viewing


One Response to “Group Theory: Unveiling The Monster”

  1. Sidney Ruthven says:

    It’s not very often I find myself interested in math but I enjoyed this article and it really got me thinking so thanks 🙂