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.