Research Track Day 1 Group Theory

Group Theory

Group Theory is the study of groups in mathematics and abstract algebra.

This is an excerpt from Cheenta Research Track training burst. Research Track program has two components.

1. Training burst (a sequence of 3/4 sessions to help students acquire necessary background knowledge). This may happen in certain months of the year.
2. Weekly / biweekly meetings (to work on a specific problem)

Group

Group is a collection of ‘forces’ that can move points in a space. (This is not definition of a group, just a way to think about it). Understanding the ‘action’ of a group on a ‘space’ helps us to understand the group better.

Groups are usually big (containing infinitely many elements). We want to break it down into smaller blocks. This is similar to factorization of large numbers into prime factors. In fact, it is a common theme all across life: see a big problem? Break it down into small, manageable parts and try to understand the parts.

How do we factorize groups?

One way is to understand group action on a space. We won’t give definitions here. Rather, we will give examples.

First example

Consider the group of integers: {0, 1, -1, 2, -2 .. }

Why is this set a group? It satisfies the four conditions that make a set a group:

1. Add any two and you will get a third element of the set (hence set of odd numbers is not a group. )
2. There is an identity element (the do-nothing element). In this set it is 0. Add it to any other element a. You will get a back.
3. Each number has inverse — the undo operation. For example element 2 has an inverse: -2
4. addition is associative

Next consider the space of real line (\( \mathbb{R}\). It is the set of real numbers.

Finally consider the action of G (group) on the S (space). Here is the catch - point. You have to image each group element as a force which can potentially move a point in the space using a certain rule.

There can be many rules. We are interested in some. They should have a couple of desirable properties:

1. The do-nothing element of the group (identity element) should literally be the 0-force and not move any element of the space at all.
2. If \( g_1, g_2 \) be two forces and P be a point in the space. Suppose applying the force \( g_1 \) to P it moves to Q. Then applying \(g_2\) to Q, it goes to R. But the action is such that you are allowed to do a different thing: combine the forces \(g_2, g_1\) inside the group and then apply the resulting force to P and you will reach R.

If you know the basic definition of group action, even then it helps to think about it in this way.

Group Theory Action

What will the force 2 do to the point 5.3?

It may send 5.3 —> 7.3. It may send 5.3 —> 3.3. We can define other weird rules as well. For example \( 5.3 —> 5.3^2 \). Somehow we have to use the numbers 5.3 and 2 and think about 5.3 as a point on the line and 2 as a force.

If the rule is translate to the right then we get the circle from the line! This was discussed in the very last section of this session (fundamental group).