What is a group?
Group theory is a subject associated with abstract algebra, a core course in undergraduate math. It deals with structures called groups, which are a collection of objects that interact with each other using an operation. Examples of groups include the set of integers, the set of real numbers, and the set of possible rotations and reflections of a square. Groups obey the following characteristics:
- Closure: Performing the operation on two elements in a group results in an element that is in the group.
- Identity: The group has an identity, i.e. performing the operation on the identity and an element in the group results in the same element.
- Inverse: For every element in the group, there exists an element called the inverse such that the operation on an element and its inverse results in the identity.
Example of a group
One great example of a group is the integers under addition (a way of saying that the group operation is addition). The identity element would just be 0 (as 0 + any number is just that number). The inverse of a number n is just -n (since n + (-n) is 0, the identity). This group has infinite elements, but other groups can have a finite number of elements. One especially interesting group is a Rubik’s Cube – there are only a set number of twists you can make on a Rubik’s Cube, and their operation is composition.
Prerequisite/What I need to know
For group theory, there are really no prerequisites other than what could be called “mathematical maturity”. This means being comfortable with reading, understanding, and writing proofs. Some set theory is also helpful, but not necessary to learn the subject. Beware: group theory is possibly one of the hardest math classes you can take. It’s not impossible to learn, but it strikes people as difficult because most people are not used to thinking about mathematics in this way. It’s very general and abstract, which makes it somewhat difficult to visualize, but the best way to become comfortable with it is to do practice problems that apply the concepts covered in these resources.
Group Theory Resources
- Socratic – Abstract Algebra (A video series that helps to explain some of the core concepts in abstract algebra – first 23 videos are on group theory)
- Contemporary Abstract Algebra by Joseph Gallian (an introductory college-level text to group theory)
- A book of Abstract Algebra by Charles Pinter (another textbook)
- Even Chen’s Napkin project (Chapters 1, 3, 10, 11)
- John Conway – Monster Group (Numberphile) (an interesting supplemental video)
- Rubik’s cube group (MIT) (A paper expanding on the idea of a rubik’s cube being a group)
- Ted-Ed intro to group theory (A fascinating video that touches on the Rubik’s cube group concept and what got me into group theory)