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.
Theorem. Each element of a group has exactly one inverse. That is, for x is an element of a group G, the only element y of G with the property that xy = yx = 1, is the element x-1.
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.
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:
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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