Examples of Groups

As was mentioned in the introduction to this section, a general theory is not of much value unless there are some interesting examples to which the theory applies. In this sense, the theory of groups is a superb theory. Let us now survey some various examples of groups which naturally present themselves in various branches of mathematics.

Examples from Arithmetic

Many of the arithmetic systems which we studied in the chapter on sets were groups, even though we did not call attention to this fact at that time.

Example 1: G = Z, with respect to the binary operation of addition. As mentioned in the last section, the identity element is 0 and the inverse of x is -x.

Example 2: G = Q, the rational numbers, with respect to the binary operation of addition. The identity element is 0 and the inverse of a/b is (-a)/b.

Example 3: G = Q - {0}, with respect to the binary operation of multiplication of rational numbers. Since the product of two nonzero rational numbers is a nonzero rational number, multiplication does, indeed, define a binary operation on Q - {0}. The identity element is the rational number 1, and the inverse of the rational number a/b 0 is b/a, since (a/b)(b/a) = 1.

Example 4: G = N, with respect to the binary operation of addition. This is not a group. Axioms G1 and G2 hold, but axiom G3 does not. For example, 1 has no inverse with respect to addition in N.

Example 5: G = Z_n with respect to the operation of addition of residue classes. The identity element is 0, since x + 0 = x + 0 = x and similarly 0 + x = x for all x Z_n. The inverse of x is -x, since x + -x = x + (-x) = 0 and similarly -x + x = 0. Note that Z_n is a finite abelian group of order n.

Example 6: G = Z^x_n with respect to the operation of multiplication of residue classes. We saw in the earlier section on congruences that multiplication of residue classes is actually a binary operation on Z^x_n - that is, the product of two elements of Z^x_n is again an element of Z^x_n. The identity element is 1. if a Z^x_n, then a has an inverse with respect to multiplication by proposition 4 of the section on congruences .Therefore, G is a group. Moreover, by the discussion in the section on congruences, Z^x_n is a finite abelian group of order (n), where (n) denotes Euler's function.

The Symmetric Groups

In the section on Historical Perspectives we introduced the symmetric group S_n in connection with the solution of equations in radicals. Recall that S_n consists of all permutations

where {i₁,i₂,...,i_n} is some rearrangement of {1,2,...,n}. It is clear that a permutation is just a function from {1,2,...,n} into {1,2,...,n} - that function defined by

(1) = i₁.

(2) = i₂.

(n) = i_n.

Moreover, since {i₁,...,i_n} is a rearrangement of {1,2,...,n}, it is clear that is one to one and onto - that is, is a bijection. Conversely, every bijection of {1,2,...,n} onto itself defines a permutation. This gives us a clue how to define the product of two permutations and . The product must be a bijection of {1,2,...,n} onto itself. Let it be the function defined by composition:

(i) = ((i)).

Thus, the product is the permutation gotten by first performing the permutation and following it by the permutation . It is easy to compute products of permutations. For example, let us compute the product of , and . First, maps 1 into 3 and maps 3 into 1. Therefore, maps 1 into 1. Since maps 2 into 2 and maps 2 into 3, the product maps 2 into 3. Similarly, maps 3 into 2. Thus .

Since the composition of functions is associative, we see that multiplication of permutations is associative. Moreover, the permutation is an identity with respect to multiplication of permutations. Finally if , then has an inverse with respect to multiplication. Indeed if we set , then it is reasonably easy to see that

= =

so that is an inverse for . Thus, we have proved that S_n is a group. Note, however that S_n is not usually abelian. For example,

By induction, it is easy to see that the order of S_n equals 1·2·3·...·n = n! (read "n factorial"). Also, its easy to see that the inverse of S_n is just the inverse of considered as a bijection from {1,2,...,n} onto itself.

The last comment suggests a generalization of the symmetric group S_n. Let A be an arbitrary set, and let S_A denote the set of bijections of A onto itself. Then S_A can be made into a group by defining multiplication in S_A via (gf)(x) = g(f(x)), for f,g S_A, x A. The identity element is the identity function on A, and the inverse of the bijection f is the function f ^-1. S_A is called the symmetric group on the set A, and the elements of S_A are called permutations of A. If A = {1,2,...,n}, then S_A = S_n.

Examples from Geometry

Example 7: Let

R² = R × R = {(x,y) | x,y R},

R³ = R × R × R = {(x,y,z) | x,y,z R}.

Then R² is called the Euclidean plane and R³ is called Euclidean 3 - space. The sets R² and R³ may be visualized as the usual plane and three-dimensional spaces of analytic geometry, respectively. The groups which we consider will consist of various motions of R² and R³. For the moment, let us confine ourselves to R².

A motion of R² is a bijection of R² onto itself; that is, a motion of R² is just a permutation of R². A motion of R² is what was classically called a "transformation." From our study of symmetric groups, we know that the set of all motions of R² forms a group under the operation of composition of functions. This group will be denoted by M(R²). Most motions of R² are uninteresting. However, many geometric situations give rise to interesting collections of motions which form groups under the operation of multiplication of permutations. It is useful to think of a motion A of R² as moving the point (x,y) to the point A((x,y)). The product of A · B of the two motions A and B moves the point (x,y) to the point A(B((x,y))) - that is, taking the product of A · B amounts to first performing the motion B, then the motion A.

Example 8: A linear motion of R² is one which moves the point (x,y) to the point (x',y'), where

and ad - bc 0. The last condition guarantees that for every (x',y') there exists a unique (x,y) which satisfies the system of equations (2). For if (x,y) satisfies (2) for a given (x',y'), then by multiplying the first equation by d and the second by b, and then subtracting the second from the first, we see that dx'-by' = (ad-bc)x, and thus, since ad-bc0, we have

Similarly,

If A' denotes the linear motion which maps (x',y') into (x'',y''), where

then A' · A maps (x,y) into (x'',y''), where

Moreover,

(a'a + b'c)(c'b + d'd)-(c'a + dc')(a'b + b'd)=(ad - bc)(a'c' - b'c')0.

Therefore, A' · A is a linear motion. The identity motion is clearly a linear motion and the inverse of the linear motion A is described by formulas (3) and (3'). Thus, the set of linear motions of R² forms a group, called the two-dimensional general linear group, denoted GL(2,R).

Example 9: A translation of R² is a motion which moves the point (x,y) to the point (x',y'), where

x'= x + a,
   y' = y + b

We can visualize a translation as moving the whole plane along the vector whose initial point is the origin and whose terminal point is (a,b) (Figure 1).

The product T₁ · T₂ of the two translations

is the translation

The identity motion is clearly a translation and the inverse of the translation

is the translation

The set of translations of R² forms a group, denoted T(R²). The translation (4) will be denoted by T_a,b.

Example 10: The motion of R² obtained from the following linear motion M by a translation T_a,b is called an affine motion of R². If M corresponds to

then the affine motion T_a,b· M moves the point (x,y) to the point (x',y'), where

x' = x + y + a,

y' = x + y + b.

The set of affine motions of R² forms a group, denoted A(R²). We will leave the verification of the group axioms to the reader. to see the results of an affine motion click here.

Thus far, all our examples of the groups of motions have been infinite. But many interesting finite groups occur as groups of motions. Let us give some examples.

Example 11: A rigid motion of R² is a motion of R² which preserves distances between points. Recall from analytic geometry that the distance between two points (x,y) and (x',y') is given by

An example of a rigid motion of R² is the motion which rotates the plane through an angle. This motion will be denoted R. Another example is the translation T_a,b.

Let S R² be any subset. A symmetry of S is a rigid motion which maps S onto itself. At first this may seem like a strange definition, but it really does coincide with our intuitive concept of symmetry. Let us consider an example. Let S be a square, centered at the origin, as in Figure 2.

The identity motion I is clearly a symmetry of the square. Let R be the motion of rotation counterclockwise though an angle /2. The square of Figure 2 is mapped onto itself situated as shown in Figure 3(a). Therefore, R is a symmetry of the square. Similarly, R · R = R² and R · R · R = R³ are symmetries. (R² is just a counterclockwise rotation through an angle of , while R³ is just a counterclockwise rotation through an angle of 3/2.) Note, however, that R⁴ brings the square back to its original position, since R⁴ is a counterclockwise rotation through 2. Therefore, R⁴ = I.

Another symmetry of the square can be gotten by rotating the square around the X-axis, resulting in the configuration of Figure 4. Let us call this symmetry F (for "flip"). Then it is clear that upon repeating F twice, the square is returned to its original position. Therefore, F² = I. Moreover, a trivial calculation shows that R · F = F · R³.

By combining flips and rotations, we get symmetries of the form

It might seem that this list should go on to include many other symmetries, say F · R³ · F. However, since R · F = FR³, R⁴ = I,, F² = I, we see that

With a little patience, one can show that the product of any two of the elements (5) is another element of the form (5), and that the elements of (5) form a group. We have assembled the multiplication for the group in Table 1. (Tables of this type are also referred to as Cayley tables)

Table 1

Multiplication Table for the Group of Symmetries of the Square

	I	R	R2	R3	F	FR	FR2	FR3
I	I	R	R2	R3	F	FR	FR2	FR3
R	R	R2	R3	I	FR3	F	FR	FR2
R2	R2	R3	I	R	FR2	FR3	F	FR
R3	R3	I	R	R2	FR	FR2	FR3	F
F	F	FR	FR2	FR3	I	R	R2	R3
FR	FR	FR2	FR3	F	R3	I	R	R2
FR2	FR2	FR3	F	FR	R2	R3	I	R
FR3	FR3	F	FR	FR2	R	R2	R3	I

If, in this table, we look up element a on the row index and b on the column index, the tabular entry gives the value of a · b. You are strongly encouraged to verify the entries in the table, using the same reasoning as used above. The table can easily be used to check that every element has an inverse. Thus, once the table is derived, it is easy to verify that the elements form a group. Note that in this case, the associativity is guaranteed, since permutations are functions, the group operations is compositions of functions and composition of functions is associative.

The group which we have constructed is the group of symmetries of the square. It is a group of order 8. There are symmetries of the square which have not yet been mentioned. For example, the symmetry gotten by rotating the square about the Y-axis; or about the line y = x; or about the line y = -x. The reader should have no problem verifying that these symmetries can be expressed in terms of products of powers of F and R, and are therefore in the group we constructed. Note that he group of symmetries of the square is nonabelian. Indeed, we have

R · F = F · R^-1 = F · R³ F · R.

Example 12: Let us generalize the preceding example. Let n > 2 be a positive integer, and let P_n be a regular polygon of n sides with one of its vertices on the X-axis.(Figure 5.)

Figure 5: P₆

This last requirement is actually unnecessary but will make our exposition easier. We will construct the group of symmetries of P_n. Two obvious symmetries are suggested by Example 11. Let R denote a counterclockwise rotation through an angle of 2/n, and let F denote a rotation about the X-axis. It is then simple to check that R and F satisfy the following relations:

Rⁿ = I,    F² = I,

R · F · R = F,

where I denotes the identity symmetry. As in Example 11, we can show that the product of any two elements

I, R, R²,..., R^n-1, F, F·R, F·R²,..., F·R^n-1

is again and element of the same form. Of course, in this example, things are slightly more complicated, so let us outline a more efficient analysis than given in Example 11. First note that since R^n-1·R = Rⁿ = I, we have R^-1 = R^n-1, where we have used the uniqueness of the inverse. Therefore, we see that the relation R·F·R = F is equivalent to R·F = F·R^-1 = F·R^n-1. Therefore, if a is any positive integer,

R^a·F = F·R^a(n-1).

Let us now consider the product of two elements F^a·R^b and F^c·R^d, where a, b, c, and d are positive integers. Without loss of generality, we may assume that

0 < a,   c < 1,    0 < b,   d < n - 1.

Then if c = 0,

F^a·R^b·F^c·R^d = F^a·(R^b·F)·R^d = F^a·F·R^b(n-1)+d
= F^a+c·R^d-b.

Therefore, we have proved closure under multiplication. We leave the verification of the remaining group axioms to the reader. The group which is defined in the preceding discussion is called the dihedral group and is denoted D_n. In particular, if n=4, D₄ is the group of symmetries of the square. Note that D_n always contains 2n elements.

Example 13: We can speak of symmetries of subsets of R³ by taking over piecemeal the definitions for R². The three-dimensional analogues of the dihedral groups are gotten by looking at the symmetries of regular solids. In high school geometry it is proved that there are only five regular solids - the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. Their groups of symmetries are very complicated.

Examples from Analysis

Example 14: Let denote the set of all functions f : RR. Define the sum of two functions f,g the be the function f + g defined by

(f + g)(x) = f(x) + g(x) (x R).

With respect to the operation +, becomes an abelian group. The identity element is the zero function 0 defined by

0(x) = 0   (x R).

The inverse of the function f is the function -f defined by

(-f)(x) = -f(x)   (x R).

Example 15: Let C denote the set of all continuous functions. Since the sum of continuous functions is a continuous function, we see that if f,g C, then f + g C. Therefore, the operation + defined in Example 14 also defines a binary operation on C. With respect to this binary operation, C becomes an abelian group.

Miscellaneous Examples

Example 16: We may put a group structure R² as follows. Define a binary operation called addition on R² setting (a,b) + (c,d) = (a+c, b+d), where a + c, b + d denote the sums of a,c and b,d as real numbers. The identity element of R² with respect to this binary operation is (0,0), and the inverse element of (a,b) is (-a,-b). The reader should have no difficulty in verifying these assertions.

Example 17: This example provides us with a method for manufacturing new groups from given ones. Let G and G' be groups, and let G × G' denote the Cartesian product of G and G'. Let us define a binary operation on G × G' by setting (g,g')·(h,h') = (gh, g'h'). Then it is easy to see that this binary operation is associative. An identity element is (1_G,1_G'), and the inverse of (g,g') is just (g^-1,g'^-1). The group G × G' is called the direct product of G and G'. If G = G' = R, then G × G' is just the group of Example 16.