The trivial group. Every group has an identity element, so the smallest possible order of a group is 1. And taken by itself, a group with only the identity element as an element is a group. It's called the trivial group.

The group of order 2. Now, let's suppose a group G has the next smallest possible order, that is, |G| = 2. Besides the identity element, there is one other element. Let's denote that other element a. Then G = {1,a}. Because the identity element times any element equals that element, we have most of the the multiplication table for G already. We just don't know what a times a is.

But a² has to be either 1 or a since those are the only two elements of G. Let's investigate what happens if a² = a. We'd would like to just cancel an a on each side of the equation to conclude a = 1. Well, cancellation does work in groups, and you can prove it. So, prove the following theorem.

Cancellation Theorem. Suppose that the equation xy = xz holds in a group. Then it follows that y = z. Likewise, yx = zx implies y = z.

Note that the cancellation theorem is equivalent to the statement that each element in the group appears exactly once in each row and in each column of the multiplication table.

We're not done yet, because we haven't shown that this multiplication table describes the binary operation for a group. In particular, we haven't verified that multiplication is associative, but that's easy to check for such a small group, and we haven't shown that G has inverses, but it does since 1^-1 equals 1 (as it always is in a group), and a^-1 equals itself a, as you can see from the multiplication table.

Thus, we've found the only group of order 2. It's got to have one element besides 1, and the square of that element has to be 1.

When the square of an element equals 1, but it doesn't equal 1 itself, we say the element has order 2. More generally, if xⁿ equals 1, but no lower power of x equals 1, then we say the element x has order n. (Here, n is a positive integer). The identity element 1 is the only element of a group with order 1. Don't confuse the order of an element in a group with the order of the group itself. They're different, but as we'll see later, they are related.

In summary, the only group of order 2 has the identity element and an element of order 2.

The group of order 3. An analysis like that we just did works to show that there is just one group G of order 3. This group has three elements—the identity and two others we can denote a and b. That analysis will lead you to the following multiplication table.

As, as before, you can show that this multiplication table is the multiplication table for a group. Two of the axioms for groups are easy to check if you know the multiplication table. You can tell that 1 is the identity element because the row in the table for 1 has the same entries as the entries at the top of the columns, and the column in the table for 1 has the same entries as the entries to the left of the rows. Also, it's easy to check that every element has an inverse because every row and column has a 1 in it. However, the most important axiom, that of associativity, is not easily checked from the table. To verify that x(yz) = (xy)z for all possible choices of x, y, and z in the group is a lot of work. Still, you can do it when the table as small as this one is.

Thus, we've found the only group G of order 3. It's got two elements besides the identity, both of order 3. We'll find more than one group of order 4.

The group G we've just examined of order 3 is abstract, that is, it's elements aren't described as a transformations of anything. But it often helps to understand a group by interpreting its elements as some kind of transformation. Note that in G, the cube of the element a is the identity. A transformation of the Euclidean plane with this property is a 120° rotation. We can interpret G = {1,a,b} as a group of three rotations about the origin—the trivial rotation 1 of 0°, the 120° rotation a (in the counterclockwise direction), and the 240° rotation b. The equation a² = b says that if you rotate the plane about the origin 120° twice, you get the same thing as if you rotate once about the origin 240°. The equation a^-1 = b says that the inverse operation to rotating the plane 120° about the origin 120° is, also, the same thing as rotating once about the origin 240°.

Groups of order 4. An detailed analysis of possible groups quickly gets complicated. Still, you can do it for groups of order 4. If you want find all the groups of slightly larger orders, you'll need to develop some theorems about the structure of groups.

So, what will you find if you do the analysis? Besides 1, there are are three other elements of a group G of order 4; you can call them a, b, and c. It will take quite a bit of work, and you'll have to consider lots of cases, but eventually you'll find two different multiplication tables for two groups G₁ and G₂, both of order 4.

These tables look different, but looks aren't enough. Here's another multiplication table

The table for G₃ looks different from the other two tables, but the group G₃ is isomorphic to the group G₂. The only difference is in the names of the elements and the order they're listed in the tables. In particular, the names of the elements b and c have been interchanged. Roughly speaking, two groups are isomorphic if they have they have the same multiplication, but the names of the elements can be different. Here's a formal definition for isomorphism:

Definition of isomorphic groups. A group G is isomorphic to a group H when there is a one-to-one correspondence f between the elements of G and the elements of H that preserves the group structure. More precisely,
1. f relates the identity element of G to the identity element of H (that is, f(1) = 1),
2. f relates inverses in G to inverses in H (that is, f(x^-1) = (f(x))^-1 for every element x of G), and
3. relates products in G to products in H (that is, f(xy) = f(x)f(y) for every pair of elements x and y of G).

This definition can be simplified since any one-to-one correspondence that relates products to products will automatically relate the identity to the identity and inverses to inverses. Since isomorphisms preserve all the group structure, they have to preserve the orders of elements—if an element x of the group G has order n, then the element y = f(x) also has to have order n, and vice versa. Consider G₁ and G₂ described above. In G₁ there are three elements of order 2, but G₂ has two elements of order 4 and only one of order 2. That's enough to conclude that G₁ and G₂ are not isomorphic groups.

There are many interpretations of G₁ and G₂ as transformation groups. Here are a couple. You can interpret G₂ as a group of rotations of the Euclidean plane about the origin, like we did for the group of order 3, where a is a 90° rotation this time, as displayed in the figure to the right, b is the square of a, a 180° rotation, and c is the cube of a, a 270° rotation.

Unlike G₂, the group G₁ can't be interpreted as a group of rotations of the Euclidean plane. It can, however, be interpreted as a group of certain rotations in Euclidean space. Its three elements a, b, and c each have order 2, so if they're interpreted as rotations, they'll each be 180° rotations. One way of doing that is to interpret a as a 180° rotation around the x-axis, b as a 180° rotation around the y-axis, and c as a 180° rotation around the z-axis. See if you can show that ba = c, that is, if you first rotate 180° around the x-axis, then 180° around the y-axis, the result is the same as if you just rotated 180° around the z-axis.

When you've got an abstract group G and you interpret it as a group of transformations of some set S, you can describe what you've done in a couple of ways. You can say you've represented G as a group of transformations. You can also say you've found a group action of G on the set S. An individual group G can be represented in many ways. Here's a different representation of G₁, this time as transformations of the Euclidean plane. Interpret the element a as a reflection across the x-axis. So a(x,y) = (x,–y). Interpret the element b as a reflection across the y-axis, so that b(x,y) = (–x,y). Then c will have to act as follows:

Thus, c is a rotation 180° about the origin. This 2-dimensional representation of G₁ is very different from the 3-dimensional representation of G₁ described above where a, b, and c were all represented as 180° rotations.

Different group representations bring out different aspects of abstract groups.

Copyright © 2003. David E. Joyce

These pages are located on the web at
http://aleph0.clarku.edu/~djoyce/modalg/