No one likes higher math on totse? Well anyway here’s something I wrote up for T2:
Before we talk about groups, we must talk about sets and binary operations. A set is a collection of well defined elements which can be finite or infinite. For example, a set S could be {1, 2, 3} or the set of all integers. A binary operation on a set S is a function mapping S x S into S. What this means is you take two elements from some set, perform some kind of operation with them, and the result is a single element from the same set.
Addition and multiplication are binary operations, but since this is Abstract Algebra, we use the symbol * to represent any binary operation. The whole idea is basically that the usual mathematical operations you’re familiar with are human constructs. * could represent anything. For example, we could define a * b to mean a – 2b.
A group is a set closed on a binary operation. This is usually denoted as (G, *), where G is the set and * is the binary operation. Groups have other properties, namely:
1. The binary operation is associative. This means for any a, b, c in a group G, a*(b*c) = (a*b)*c. You’re really just shifting the parenthesis. This is often taken for granted, but consider a * b to mean a ^ b. Now, let a = 2, b = 3, and c = 4. 2 * (3 * 4) = 2 ^ (3 ^ 4) = 2 ^ 81, but (2 * 3) * 4 = (2 ^ 3) ^ 4 = 8 ^ 4. The results are not equal, so * is not associative in this example.
2. The group G contains an identity element. In G, there must exist some element, usually denoted as e, such that a * e = e * a = a for any a in G. In other words, there must exist a unique element in the set that when you take that element and any other element a, the result is a under the binary operation. The identity element must work on both sides of a (communative). For addition, the identity element is usually zero, i.e. zero added to anything will yield the original value. For multiplication, it is usually one.
3. The group G must contain inverses. For any a in G, there must be an element, usually represented as a’, such that a * a’ = a’ * a = e. In other words, every element must have some kind of compliment in the set that yields the identity element under the binary operation. This inverse of a must work on both sides of a. For addition, it’s a and -a. For example, 5 + -5 = 0, which is the identity element in integer addition. For multiplication, it’s a and 1/a. For example, 7 x 1/7 = 1, the identity element in multiplication.
Now let’s think of some group examples. Consider the integers and addition. Addition is associative, zero acts the identity element, and each integer a has an inverse in the form -a, so it is a group. However, think of the integers under multiplication. Multiplication is associative, one acts the identity element, but what about inverses? The number 10 is an integer, and 1/10 is its multiplicative inverse. But 1/10 isn’t an integer, so this isn’t a group because there’s no inverses. What if we extended our multiplication example to all real numbers? Each real number a would have an inverse in the form 1/a. But zero is a real number, and there’s no real number you can multiply by zero and get the multiplicative identity, one. Thus, only the real numbers excluding zero is a group under multiplication.
Having fun yet?
