What is a mathematical group? (Part 1)

Last semester, a friend of mine (who has now graduated with his B.S. in physics) asked me the question, “What exactly is a group?”

I was able to answer his question because I had already taken a course in elementary group theory. But I had also taken a course in particle physics, in which the Standard Model relies on groups, and I had not heard anything about defining a group in this class. For the uninformed student, it seemed like the lecturer was doing weird magic with groups and “adding” them together.

I will be writing this short introduction with the physics student in mind. Especially if the student is interested in any type of particle physics, this may be an interesting read.

To begin on the odyssey that is in store for a student of abstract algebra, it pays to know a bit of terminology.

Definition: A binary operation (sometimes law of composition) on a set is a function . That is, this function takes two elements of the set and produces another in . A more familiar way of looking at this for a physicist that has taken a course in linear algebra is that the set is closed under this operation, just as in a vector space – the sum of two vectors in a vector space is another vector in that same set.

For example, addition (and subtraction as well) is a binary operation on the set of all integers , for if one adds (or subtracts) two integers, the result is another integer.

Typically, this binary operation is not written in the usual fashion of writing a function of multiple variables , but rather . Normally, unless it leads to confusion, the is dropped in favor of just writing as in normal multiplication.

Definition: The binary operation is associative if for every in the set on which the operation is defined, . That is, the order of applying the binary operation has no effect on the outcome.

For example, if we examine the example of integers with addition as the binary operation, this binary operation is associative. However, if one takes the operation as subtraction, this is not associative (take, for example, 1 – (2 – 3) = 1 – (-1) = 2, but (1 – 2) – 3 = -1 – 3 = -4).

Definition: An identity element of a set with a binary operation is an element where for any .

Note: Mathematicians like to use shorthand (hence all of the symbols used for different operations). The symbol is read “is an element of”. So the previous line is read “…for any that is an element of .”

Going back to the example of integers with addition, 0 acts as the identity here, because for every .

Definition: Given an element in a set with a binary operation and identity element , the inverse of (usually denoted by where appropriate) is one such that .

One last reference to the integers under addition shows that the inverse of an integer is since .

The punchline from all of this is the definition of a group:

Definition: A group is a set together with a binary operation that has the property of associativity, an identity element, and inverses for each element in the set.

Next time, I’ll give some examples of groups and start proving some elementary theorems about them.

(Part 2)(Part 3)