Book recommendations for learning group theory

+ some video recommendations as well!

What is group theory?

Take a look at these two shapes, a letter A and a square:

Intuitively speaking, they are both “symmetrical”. But what does this mean precisely?

• The letter A is symmetrical because if you flip the letter A across the vertical axis, it looks the same.

• The square is symmetrical because if you rotate the square by 90-degree increments in any direction, it looks the same.

In other words, when we say that a shape is “symmetrical”, we mean that there are actions you can perform on it that leave them looking the same.

But the square is more symmetrical than the letter A. This is because there are more actions you can perform on the square that leave it looking the same.

The set of all actions you can perform on an object that leaves it looking the same forms something called a group. The mathematical study of groups is called group theory.

(Disclaimer: This is not at a mathematically precise definition of a group. See the resources I’ve linked below for more details.)

What is group theory good for?

A recurring theme in modern math is:

To study an object, study its symmetries.

There are so many examples of this, so I’ll mention one here: the insolvability of the quintic. This famous theorem says that the roots of a polynomial of degree >4 cannot in general be expressed in radicals.

This proof studies the symmetry group of the roots of that polynomial. It shows that every polynomial whose roots are expressible in radicals has a symmetry group must be a so-called “solvable group”. The proof then shows that not every degree >4 polynomial has a symmetry group which is solvable. Therefore, its roots cannot be expressible in radicals.

This is an example of a problem that can be solved using group theory which, crucially, does not require group theory to state.

How do I learn group theory?

Video Recommendations

1. Videos by Socratica about abstract algebra.

These videos are an absolute gem. They are in bite-sized 5-minute chunks and they motivate all the concepts very clearly.

2. Online lectures by Prof. Benedict Gross.

When I was learning abstract algebra in undergrad, I watched these lecture videos religiously. They start off by explaining the basic definitions of group theory and by the end, they arrive at the more difficult theorems in the area.

Book Recommendations

If you're more of a books person, here are some books that I found helpful:

1. Contemporary Abstract Algebra by Gallian

2. Topics in Algebra by Herstein (Chapter 2)

The first book here is very friendly. It has tons of historical insights as well as worked examples and exercises. Herstein’s book is also very good but it is more advanced so I’d recommend looking into it as a second pass on the subject.

Challenge Problem

As always, let’s start off this issue with a challenge problem. Here it is:

The product of three positive reals is 1. Their sum is greater than the sum of their reciprocals. Prove that exactly one of these numbers is strictly greater than 1.

Got a solution to the challenge problem? Submit it here.

Solution from last week

See here for the solution to last week’s problem. (Special thanks to Willow, whose solution is being featured this week.)

Thanks for reading and happy learning! Until next time,

Adithya