Group Attributes

There are a few things we need to talk about in regards to groups before we can start proving their properties. Since so many things are groups, we like to classify them in some different ways to make life a little easier.

Definition: Abelian
Let G be a group. G is said to be abelian if for each a and b in G, a∙b = b∙a.

An abelian group has the property of commutivity. Notice that it says for each, so a∙b = b∙a must hold for every single element of the group - not just some. Addition and multiplication is commutative, so the integers under addition and the rationals under multiplication are both abelian. It is not hard to see that the cyclic groups are abelian. However, the dihedral group is not abelian. Most of the "interesting" groups are non-abelian.

Definition: Order of a Group
Let G be a group. The order of G, denoted by |G|, is the number of elements in the group.

This is a pretty boring definition. All you have to do is count the elements in the group. Z(n) has order n, the dihedral group of order 8 has, well, order 8, and the integers under addition have infinite order. This brings us to our next definition.

Definition: Finite
A group is said to be finite if its order is finite. Otherwise, it is said to be infinite.

This is another boring definition, but it is important to be able to talk about a group being finite or infinite and to talk about the order of a group, so there are the definitions.

Before we can move on there are a couple of notational issues that we need to discuss. Some groups we call "multiplicative" and some groups we call "additive." All this means is that we use the ∙ or + symbol (respectively) to represent whatever the operation of the group might be, even if that operation is not multiplication or addition. It doesn't really make a difference but sometimes its just convenient to use one or the other. From now on if G is an arbitrary group, I will assume that it is multiplicative in my notation. However, this choice is basically arbitrary and does not have anything to do with multiplication.

Now, if G is an additive group, g is an element of G, and k is an integer, we use k∙g (or sometimes kg) to mean g+g+...+g a total of k times, or the sum of k copies of g. Similarly, in a multiplicative group, exponential notation is used - that is if g is an element of a group and k is an integer then g^k is g∙g∙...∙g a total of k times, or the product of k copies of g. We then get the following definition.

Definition: Order of an Element
Let G be a group, g be an element of G, and e be the identity element of G. Then suppose k is the smallest positive integer such that k^g = e. We call k the order of g and denote it by |g|. If there is no such integer, we say that |g| is infinite.

Whereas the order of a group was kind of a boring definition, the order of an element is actually pretty cool. Lets talk about a couple examples. Take 4 in Z(6). 1∙4 = 4, 2∙4 = 4+4 = 2, and 3∙4 = 4+4+4 = 0, so |4| = 3. Now, in the dihedral group of order 8, lets call the 90 degree clockwise rotation, R. R² = R∙R is a rotation of 90 degrees twice, so is the rotation by 180 degrees. Then R³ = R∙R∙R is a rotation of 90 more degrees, or the rotation by 270 degrees. Finally, R⁴ = R∙R∙R∙R is a rotation by 360 degrees, which is the same as doing nothing, so |R| = 4.

Definition: Cyclic
Let G be a group. Then G is cyclic if there exists an a in G such that for each g in G there is an integer, k, such that a^k = g. In this case, we use the notation G = <a>.

This definition is why I called Z(n) the cyclic group of order n. If k is in Z(n) then k∙1 = k so Z(n) = <1> This might be confusing, because I used k both times when I say k∙1 = k. The first k is the required integer and the second k is the group element. Also note that I'm not using multiplicative notation - this is because the operation on the cyclic groups acts very much like addition. The notion of a cyclic group is a hard definition to explain because it just is what it is - sometimes groups satisfy the property and sometimes they don't.

We've now worked our way up to a point where we can finally start doing some real work with groups and we can start talking about some interesting stuff. For reference, I've now covered the first two or three weeks of an introductory course in group theory designed for aspiring math majors - what we're working toward here is some pretty advanced stuff. Next time I'm going to start doing some actual proofs and getting some concrete results.

References

Previous Related Post: Examples of Groups

Text Reference: Gallain Chapter 2

Wikipedia: Group