Post #1: The definition of a group

 Welcome to my new blog, which will study the mathematical field of group theory. I know it will have a limited audience, and it won't be exactly like an online textbook, because I will include commentary about my own history studying the field, these comments will usually be saved until the end of the post.

 

Definition of a group

A Group G=(S,*) is a non-empty set S and a binary operator *, which works on the elements of S and follows these rules.

1. The group is closed under the operation, meaning if a and b are elements of G, a*b is also an element of G.

2. There is a unique identity element, and in this definition we will call it e. For any element a, a*e = e*a = a.

3. For every element a, there is an inverse element a⁻¹ such that a*a⁻¹ = a⁻¹*a = e, the identity.

4. The operation * is associative, such that for any a, b and c that are elements of G, (a*b)*c = a*(b*c).

 

Examples of groups from basic mathematics

Addition (+) is a binary operator, and so is multiplication (✕). These are the operators we will use most often in our early examples, but other operators exist, as we shall explore in later posts. Subtraction and division, are not usually used as operators in groups, because they are not associative. Here are two simple examples.

 

(2-3)-4 = -1-4 = -5, but 2-(3-4) = 2 - (-1) = 3

(2/3)/4 = 2/12 = 1/6, but 2/(3/4) = 8/3

Now let's define some sets of numbers, which we will symbolize with bold italics letters from the alphabet.

 

Z: the integers {..., -2, -1, 0, 1, 2, ...}

Q: the rational numbers, p/q, where p can be any integer and q is an integer greater than 0.

R: the real numbers, which means we include the irrational numbers

C: the complex numbers, all numbers of the form a+bi, where i represents the square root of -1, often called the imaginary number.

 

Additive groups

(Z,+), (Q,+), (R,+) and (C,+) are all additive groups. The identity in an additive group is 0, and the inverse of a is -a.

 

Multiplicative groups

When multiplying, 0✕a = a✕0 = 0 for all a. This means 0 does not have a multiplicative inverse, because the multiplicative identity is 1. For any number a other than 0, the inverse can be written as 1/a.

 

(Q-{0},✕), (R-{0},✕) and (C-{0},✕) are all multiplicative groups, but (Z-{0},✕) is not, because unless the integer is 1 or -1, 1/a is not an integer, so not every element has an inverse.

 

Our first finite groups

Note that every group defined so far has an infinite number of elements. There are finite groups as well, ironically infinitely many, but in basic mathematics, we see only a few.

 

The smallest possible groups have exactly one element, the identity, so ({0},+) and ({1},✕) both count as groups. The order of a group is the number of elements, so these groups are of order 1.

In the explanation of why (Z-{0},✕) is not a group, we note that 1 and -1 have multiplicative inverses. In fact, they are there own inverses, 1/1 = 1 and 1/(-1) = -1. Since (-1)✕(-1) = 1, the set {1,-1} is closed under multiplication, and multiplication is associative using any numbers we have met, so ({1, -1},✕) is our first finite group of order 2.

 

Commentary

What I have written so far is first few minutes of an introductory class. When I was in college in the 1970s, the class was called Modern Algebra, a bit of a misnomer because group theory's beginnings are from the 19th Century. To be technical, most math in the undergraduate curriculum is much older than that, but now the class is called Abstract Algebra. It is still a junior level class, which means most of the people taking it will be math majors. When I took it, I was a computer science major, but I was so enchanted by group theory, I switched to math before the end of the quarter.

Something I didn't learn in that first class is that group theory is the study of symmetry. Symmetry is not only beautiful, it is vital in solving math problems in countless fields. To consider a non-numeric example, solving a Rubik's Cube is a problem in group theory, where the basic operation is twisting a face of the cube some multiple of 90° and the set is any legal position of the cube. Because it is a physical object, associativity becomes a moot point, if the instructions are move a, then move b, then move c, we are bound to do them in the specific order.

 

Another thing I didn't learn until graduate school is that symmetry is almost irreplaceable in the study of differential equations. The only solved differential equation I know of that has no symmetry is the description of the soliton, or solitary wave. The most famous physical representation of the soliton is a tsunami, though they can be much smaller, and when smaller, they aren't particularly destructive.

 

Tomorrow: Groups and subgroups