Algebra Topics: Defining the Mathematical Group

Have you ever thought about alien cultures? Have you thought about how different their societies might be? Their technology? Their languages? Well, one thing we would know about an alien culture: their mathematics. How would we know that, though? The answer is the subject of abstract algebra.

What is abstract algebra? Basically, abstract algebra is an area of mathematics that defines mathematical structures in general terms. The purpose of this is to generalize the functioning of mathematics.

For example, most modern cultures and any culture that has international dealings in business use a system of integers with 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We all know that 1 + 9 = 10. Well, truthfully, that's not always the case!

We commonly use a base 60 system in dealings with time. 10 seconds is not 1 minute, right? No, 60 seconds equals 1 minute. Computers use base 2 systems (binary) for bits. For example 1 + 1 = 10 and 10 + 11 = 101. Also, computers use base 16 systems (hexadecimal), in which 9 + 1 = A.

What does this have to do with Algebra? How can we predict the behavior of these types of systems? The answer to this valid question is the Group.

Let's take a set of elements. For sake of familiarity, let's use the collection of integers. Now, let's take the most common operation "+" that we have used since the first day of first grade. Consider any two elements of our set, say 8 and 5. Now, 8 + 5 = 13.

It is easy to see now that by adding any two elements in our collection, the sum will also be in our collection. In general terms that is a + b = c, where c is in our set. This property is known as closure.

Also notice that addition in this context is associative. That is, a+(b+c) = (a+b)+c.

Now, our collection has 0. We all know that x + 0 = x. This 0 is known as the identity for our operation, "+". Mathematically, we call the identity e, where a+e = e+a = a.

Let's ponder e (0) for a moment. What two elements add to equal 0? Well, x + (-x) = 0. Now, if x is an integer, -x is also an integer, so for any element x, there exists an inverse -x. Mathematically, we say for any a, there exists a^-1 where a + a^-1 = e.

We've concluded that the collection of integers is closed under "+". It is associative. It contains an identity, and every element has an inverse. These terms happen to be the rules that define a group.

Quite simply, the Group G is a set closed under a binary operation * denoted by (G, *)in a way that the operation satisfies three rules:

1. The operation is associative a*(b*c) = (a*b)*c

2. The operation contains an identity element, say e where e*a= a

3. Each element in the Group contains an inverse a^-1 where a^-1 *a=e

Let's talk about this idea for a minute. Why is it important if the collection of integers is a group with addition? Well if we can think of a collection of elements that also follow these rules, then we know that this is a group. So, basically the rules that apply to addition of integers also apply to these other groups.

Notice that the integers with multiplication is not a group! Why not? First, think of the multiplicative identity, 1. Now, for any element a, what is its inverse? 1/a, which is not an integer. Therefore, there are no inverses.

Ok, so lets say the collection of rational numbers with multiplication. Sorry, 0 has no multiplicative inverse. However, the collection of rational numbers without 0 with multiplication is a group! So this group basically operates in a similar manner as the integers with addition.

Let's take a look of a collection of matrices with dimensions n X n, real number entries, and inverses. Also, lets let * be matrix multiplication. After a few calculations, we can see that this set is closed and associative. There is an identity matrix and also inverses by definition. So by definition, this collection is a group!

However, consider two matrices A and B. After a simple calculation, we can see that A * B does not equal B * A. So this group is non-commutative.

If a group is commutative, it is known as an abelian group.

So the group helps us view seemingly unrelated objects as similar through mathematics. If we can find these rules and an operation, we can compare it to more familiar groups. In the case of an alien culture, even though the details of their mathematics may be different, the generality will be the same. In fact, the governmental agency SETI hopes to contact extraterrestrial cultures through mathematics via similar methods as we have discussed.