Topics of Abstract Algebra II: Group Examples I

I already mentioned the formal definition of an Algebraic Group, but I didn't go too into detail on examples. Truthfully, there are some really quite amazing examples of groups, and I want to mention a few of them.

We already mentioned the group (Z, +), which is the group consisting of integers with addition as its operation. You might recall, also, that multiplication cannot define Z as a group, because there are no inverses. Example: 9 * ? = 1. We know this to be 1/9, but 1/9 is not an integer.

We also mentioned the group (Q, *): the group of rational numbers without zero and has multiplication as its operation.

Let's talk about some other examples. I feel obligated to mention some of the basic examples such as (Q, +), (R, +), and (C, +), where R denotes the set of Real Numbers and C denotes the set of Complex Numbers. It is easy to see that these all hold group properties.

Take a look at this, though,

(C,+) > (R,+) > (Q, +) > (Z,+). This means that (C,+) has subgroupss which are (R, +), (Q,+), and (Z,+). Also, (R,+) has the subgroups (Q,+) and (Z,+). We haven't defined a subgroup yet, but it is basically a group contained within a group. The subgroup inherets its elements and operation from its parent group. Example: 6/5 is a rational number, but it also a real number and a complex number.

Think for a second, does (Z, +) have any subgroups? What if I told you it had an infinite number of them? Take any number (we'll say 3). If I take the set {3a | a in an integer}, I have a group. Not convinced? We know that 3a will always be an integer. We know that is it associative, and each element has an inverse for the identity 3(0) = 0. Is it closed?

Well, take a and b which are integers. Obviously 3(a) and 3(b) are in our set. Consider 3(a) + 3(b) = 3(a + b). Now, a + b is an integer, so 3(a + b) is also in our set. We can see now that our set is a group. It is also easy to see that we could have done the same thing with any number composite or prime.

A group such as this is called a cyclic subgroup. In particular, we have the cyclic subgroup generated by 3.

Ok, we discussed (Q, *). What about (C, *)? Is it a group? Associativity and identity is obvious, but what about inverses and closure?

Let's talk about closure first, take two comlex numbers A and B, where A = a + bi and B = c + di. Now, AB = (a + bi)(c + di) = ac + bci + adi + bdi². Remember i = sqrt(-1), so i² = -1. Now we have AB = ac -bd + bci + adi = (ac - bd) + (bc + ad)i, which is a complex number. So C is closed with *.

What about inverses? I'll cheat and just tell you that for complex number a + bi. Its inverse is (a - bi)/(a² + b²).

So (C, *) is a group. I think its pretty obvious that (Q, *) < (R, *) < (C, *).

These are kind of dull examples of groups. I'll mention some more exciting examples shortly.