Modern Algebra

Prof. D Joyce
Department of Mathematics and Computer Science
Clark University

Abstract groups

Abstract groups vs. transformation groups. We started out looking at transformation groups because they are more concrete, and, therefore, they are fairly easy to understand. Each element of a transformation group is a transformation on a particular set, that is, a function on the set to itself. Recall that we defined a transformation group to consist of a set G of tranformations on some set S that satisfies the following axioms.
  1. G is closed under composition: if T and U both belong to G, then so does the composition T°U.
  2. The identity transformation I belongs to G.
  3. G is closed under inversion: if T belongs to G, then so does G-1.
An abstract group, usually just called a group, is intendeed to be just the set G without the set S to act upon. That is, it's a disembodied set of transformations that might be acting on some set S, but you don't think about S. Actually, G might act as a set of transformations on various different sets, acting on each in a different way.

We need some axioms for an abstract group. First of all, there needs to be some binary operation, an abstraction of composition. Since the elements of an abstract group don't have to actually be transformations, and the operation doesn't have to actually be composition, we'll call it something else. Usually it's called multiplication, but in some applications it might be addition or something else. We'll call it multiplication here.

Now, composition is always associative, so we'll include an axiom stating that our binary operation of multiplication is associative.

We also need an identity element (which will be the identity transformation when we have a transformation group), and we'll need inverses.

All these requirements boil down to the following formal definition.

There are a lot of logical consequences of these definitions. For instance, you can easily show that there is at most one identity element of G. Since the axioms explicitly state there is at least one identity element, that means there is exactly one identity element. Also, an element x of G has at most one inverse. Try showing these directly from the axioms for groups.

Some examples besides transformation groups. Although every group can be construed as a transformation group (we'll see how later), many are as easily understood as abstract groups. Here are a few basic examples.

The integers Z. An integer is a whole number, either positive, negative, or zero. The set of integers along with the binary operation of addition forms a group. Since the binary operation is addition, we'll use additive notation rather than multiplicative notation. The identity element is denoted 0 rather than 1, and inverses are denoted as negations (–x) rather than with exponents of –1.

There are a lot of other groups with addition as their binary operations. For these groups the binary operation is commutative as well as associative, that is, x + y = y + x for any two elements x and y of the group. When the binary operation is commutative, we say the group is an Abelian group in honor of Niels Henrik Abel (1802-1829). Such groups are sometimes simply called commutative groups.

Another example of a commutative group is the unit circle S1. It can be viewed as the set of complex numbers whose absolute values are all equal to 1. (See Dave's Short Course on Complex Numbers for a refresher on complex arithmetic.) Interpreted this way, its binary operation is multiplication. For instance i5 = i. (By the way, there is no way to make the unit sphere S2 into a group, at least if you want the binary operation to be continuous. The proof that there is no group structure on the sphere is not easy. Surprizingly, the 3-sphere S3 does have a group structure.)

The general linear group GLn(R). We saw that the set of n by n nonsingular matrices with coefficients in the real numbers R was a transformation group on the n-dimensional vector space Rn, but it can just as easily be thought of abstractly as a bunch of square matrices.

Sometimes, groups are presented by "generators and relations." We'll look at that in detail later, but this is a good place to mention it. For an example, consider this presentation:

G = (x, y: x2 = y3).

This says that two of the elements of G are x and y. Other elements are formed by inverses and products of the "generating elements." For instance, y-2xy3x-1 is one such other element. The presentation also states that the square of x equals the cube of y. That implies that the element just mentioned could also be written as y-2x2 since xy3x-1 = xx2x-1 = x2. One of the difficulties with groups presented by generators and relations is it's hard to determine when two elements are equal. For instance, with this one, can you tell if xy equal to yx? (It's not.)

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Copyright © 2003. David E. Joyce

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