Group theory abstracts this one level. Instead of looking and points in the plane and distance between points, group theory starts with transformations of the plane that preserve distance, then studies the operations on tranformations and relations between these transformations. The basic set for group theory is not the set of points in the plane, but the set of transformations on the plane.

A transformation that preserves distance is usually called an isometry. An isometry T of the Euclidean plane associates to each point a of the plane a point Ta of the plane (that is, it's a function from the plane to itself), and this association preserves distance in the sense that if a and b are two points of the plane, then the distance, d(Ta,Tb), between the points Ta and Tb equals the distance d(a,b) between the points a and b. Isometries are particularly important, because if a transformation preserves distance, then it will automatically preserve other geometric quantities like area and angle. Thus, it will send a triangle to a congruent triangle.

There are several different kinds of isometries of the plane. There are translations, rotations, reflections, and glide reflections. See the my web site on Wallpaper Groups at http://www.clarku.edu/~djoyce/wallpaper/. Read, in particular, the section therein entitled "Transformations of the plane." There, you will see the illustration to the right which shows a rotation about a point by 90° along with these other kinds of isometries.

The set of all isometries of the plane has a particularly nice structure, the structure of what mathematicians call a group. There are lots of other collections of transformations that have this same structure, and any one of them would do as an example of a group, but we'll stick with this example of the isometries of the plane to illustrate this concept.

Properties of a transformation group. First, composition. Not every collection of transformations will form a "group" of transformations. To be a group, the collection will have to have some minimal properties. The first one is that is closed under composition. If T is one transformation in the group, and U is another, then you could first perform T, and then perform U. The result is that you will have performed the composition of T followed by U, often denoted U°T.

Consider our example of isometries of the plane. Suppose that T is the transformation that translates a point a one unit to the right. In terms of a coordinate system, T will translate the point a = (a₁,a₂) to the point Ta = (a₁+1,a₂). Suppose also that U is another transformation, one that reflects a point across the diagonal line y = x. Then U(a₁,a₂) = (a₂,a₁). The composition, U°T will first move a one unit right, then reflect it across the diagonal line y = x, so that

Geometrically, it's difficult to see just what kind of isometry this composition is. (It's not a translation, rotation, or reflection.) But it is clear that it's an isometry, that is, that it preserves distance. Since T and U both preserve distance, so does their composition U°T. Thus, the collection of isometries is closed under composition. That will our the first property that all groups will have.

The identity element of a group, and inverses. For each isometry T of the plane, there is an inverse isometry T^-1 which undoes what T does. For instance, if T translates the plane one unit right, then T^-1 translates it back to the left one unit; if T is a 45°-rotation clockwise about a fixed point, then T^-1 rotates the plane 45° counterclockwise about the same point; and if T is a reflection across a straight line, then T^-1 is t itself, since reflecting twice brings all the points back to where they started.

The easiest way to characterize the inverse T^-1 of a transformation T is to say that their composition is the trivial transformation that does nothing. This transformation that does nothing is called the identity transformation, and we'll denote it here as I. Thus, for any point a in the plane, Ia = a. The inverse T^-1 of a transformation T is characterized by the two equations

That the identity transformation I does nothing can be characterized in terms of composition by saying that when it's composed with any other transformation, the other transformation is all that results. That is, or each transformation T,

One more property: associativity. The remaining property of groups, associativity, is obvious for transformation groups. It says that if you have three transformations T, U, and V, then the triple composition V°U°T can be found in either of two ways in terms of ordinary composition, either (1) compose V with the result of composing U with T, or (2) compose V°U (which is the result of composing V with U) with T. In other words, composition satisfies the associative identity

Composition is always an associative property.

Usually transformation groups aren't commutative. That is, don't expect that

For instance, with the example transformations T and U above, where T is the translation to the right by one unit, Ta = (a₁+1,a₂), and U is the reflection across the diagonal line y = x so that U(a₁,a₂) = (a₂,a₁), we found that the composition U°T was given by the formula (U°T)(a₁,a₂) = (a₂,a₁+1), but you can show that the reverse composition T°U is given by the formula (T°U)(a₁,a₂) = (a₂+1,a₁). These aren't equal, so U°T does not equal T°U.

Summary of transformation groups. The group of isometries of the Euclidean plane is an example of a transformation group. In general, a transformtion group consists of a set G of tranformations on some set S, that is, functions from the set S to itself, with 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 T^-1.

Some more transformation groups. One of the most important transformation groups is the symmetric group S_n on n elements. We'll look at it later in more detail, but it's good to mention it here as an example. Fix a finite number n and a set S of n elements, S = {1,2,3,...,n}. Let G be the group of all permutations of S. That is to say, an element of G is a one-to-one correspondence of the set S to itself. This G satisfies the three axioms mentioned above since (1) the composition of two one-to-one correspondences is another one-to-one correspondence, (2) the identity function is a one-to-one correspondence, and (3) a one-to-one correspondence has an inverse which is a one-to-one correspondence. This group G of all permatations on a set of n elements is called the symmetric group on n elements, and it is denoted S_n. Note that there are n! (n factorial) elements of S_n.

There are many more geometric examples besides the group of isometries of the Euclidean plane mentioned above. A couple of natural modifications to its definition lead to other transformation groups. For example, the dimension can be changed from 2 to 3 or some other dimension. The Euclidean plane could be replaced by a hyperbolic plane, or a sphere, or a cylinder. Instead of requiring that the transformations preserve distance, that is, that they be isometries, we could require that they preserve area (or volume in dimension 3), or preserve straight lines (that is, the image of a straight line has to be a straight line, but not necessariy a straight line of the same length), or preserve some more esoteric geometric property such as orientation. Each of these modifications leads to an important transformation group.

One of the more important of these transformation group is the group of linear transformations of n-dimensional space. Fix a finite number n, fix an n-dimensional Euclidean space, and fix a point in that space to call the origin. Although it's not necessary, it simplifies things to have a fixed coordinate system for the space, too. So our space is the n-dimensional vector space Rⁿ, where every point a has n real coordinates:

As described in any course in linear algebra, a linear transformation T: Rⁿ -> Rⁿ is determined by an n by n matrix A where T(a) = b if and only if Aa^t = b^t, where a^t stands the column matrix which is the transpose of the row matrix a. (We take the transpose so we can write the transformation to the left of the vector. If you don't mind putting the transformation to the right of the vector, then you don't have to use transposes.) Then composition of linear transformations corresponds to multiplication of matrices. Just as composition is associative, so is matrix multiplication associative. The identity transformation corresponds to the identity matrix I with zeros everywhere except ones down the main diagonal. However, not every linear transformation has an inverse; for instance, projections don't. So, in order to have inverses, we only consider invertible transformations. They correspond to nonsingular matrices, that is, matrices with nonzero determinant. Thus, the nonsingular n by n matrices form a group of transformations on the vector space Rⁿ. This group is called a general linear group, and it is denoted GL_n(R). We'll study the general linear group and its subgroups in greater detail later.

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Copyright © 2003. David E. Joyce


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