6.2.1 Basics of a Group

The idea of a symmetry is fundamental in science and its in‐depth study requires the mathematical machinery of the algebraic group. Group theory is one of the most useful mathematical tools for mathematicians and scientists alike, including particle physicists, where group theory is the language for understanding the structure of particle physics.

The most fundamental component of a group is its binary operation.

6.2.2 Binary Operations and the Group

A binary operation on a set A is a rule, which assigns to each pair of elements of A a unique element of A. Thus, a binary operation is a function f : A × A → A. Two common binary operations familiar to the reader are +, × which assign the sum a + b ∈ ℝ and product a × b ∈ ℝ to a pair (a, b) ∈ ℝ × ℝ of real numbers. We now give a formal definition of a group.

Often, it happens that a * b = b * a for all a, b ∈ G. When this happens, the group is called a commutative (or Abelian) group. We also denote the group operation a * b simply as ab, or maybe by ⊕ if the group operation is addition or resembles addition. A group is called finite if it contains a finite number of elements and the number of elements in the group is called the order of the group. If the order of a group G is n, we denote this by writing |G| = n. If the group is not of finite order, we say it is of infinite order.

Table 6.3 shows some binary operations of different sets.

6.2.3 Cayley Table

The binary operation of a finite group can sometimes be illustrated by means of a Cayley table as drawn in Figure 6.16, which shows the products g_ig_j of two members g_i and g_j of the group. It is much like the addition or multiplication tables the reader studied as a child. A Cayley table is an example of a Latin square, meaning that every element of the group occurs once and exactly once in every row and column. We examine the Cayley table to learn about the inner workings of a group.

The graphs demonstrated in Figure 2 are the only graphs of order 2 and 3, respectively. For order 4 however, there are two possible groups, one is the Klein four‐group and the other the cyclic group of order 4. The group in Example 3 is the cyclic group of order 4. The Klein four‐group will come later.

In general, there is no quick way to verify the associative property like there is for the commutative property. One must check all possible arrangements to verify the property. On the other hand, if one instance where the associative property fails, then the binary operation * is not associative.

Example 3 illustrates one of the two groups of order 4. We now present the other group of order 4, the Klein four‐group.

Group	Elements	Operation	Identity	Inverse
ℤ	n ∈ ℤ	Addition	0	−n
ℚ +		Multiplication	1	n/m
ℤn	k ∈ {0, 1, 2, … , n − 1}	Addition mod n	0	n − k
ℝ − {0}	x nonzero real number	Multiplication	1	1/x
ℝ2	(a, b) ∈ ℝ2	Vector addition	(0, 0)	(−a, − b)

6.2.4 Cyclic Groups: Modular Arithmetic

The most common and most simple of all groups are the cyclic groups, which are well‐known to every child who has learned to keep time.

For example, the three rotational symmetries {e, R₁₂₀, R₂₄₀} of an equilateral triangle form a cyclic group ℤ₃ with generator g = R₁₂₀ since .

Cyclic groups also describe modular (or clock) arithmetic, which is the arithmetic we perform when keeping time on 12‐hour clock. The 12‐hour clock leads us to the cyclic group Z₁₂ with elements

and group operation

which is basically the arithmetic you do when keeping time. The “mod 12” simply refers to computing a ⊕ b by computing the ordinary sum (a + b), then taking its remainder after dividing by 12. We denote the group operation by ⊕ to remind us that the operation is addition, only reduced modulo 12. In clock arithmetic, the equation 2 = (9 + 5) mod  12, which can be interpreted as meaning five hours after 9 p.m. is 2 a.m.

The 12 hours of the clock, 0 through 11, along with the binary operation of addition modulo 12 is an Abelian group of order 12 called the cyclic group of order 12, denoted by (ℤ₁₂, ⊕). The Cayley table for this group is shown in Figure 6.20.

Figure 6.21 shows various clocks that give rise to different cyclic groups.

Note that the group is Abelian and that 3⁻¹ = 7, 7⁻¹ = 3 since 3 × 7 = 1.

6.2.5 Isomorphic Groups: Groups that are the Same

Sometimes groups appear different when looking at their Cayley tables, but after relabeling their elements, one discovers they are the same group. For example, consider two groups illustrated in Figure 6.23.

The group at the left is the cyclic group ℤ₄ = {0, 1, 2, 3} and the group at the right consists of the four numbers G = {1, i, − 1, − i} which lie on the unit circle in the complex plane, where the group operation is multiplication (see Figure 6.24).

Looking carefully at the two Cayley tables in Table 6.4, you will see that the numbers 0, 1, 2, 3 in the table for ℤ₄ are arranged in the same order as the numbers 1, i, − 1, − i in the table for G. If we make the correspondences

we see that the groups ℤ₄ and G are the same, they simply use different symbols. When two groups are the same, but only different in their symbols used in their description, the groups are called isomorphic. In this example, the described one‐to‐one correspondence is called an isomorphism between the groups. This motivates the following formal definition of an isomorphism.

Roughly speaking, for isomorphic groups, the mapping and the group operations can be carried out in either order. You can operate then map T(a * b), or you can map and then operate, T(a) ⊕ T(b), the results are the same.

6.2.6 Dihedral Groups: Symmetries of Regular Polygons

In Section 6.1, we saw how the symmetries of a figure form an arithmetic system where one can “multiply” symmetries much like we multiply numbers. The set of symmetries of an object along with the arithmetic operation of composition of symmetries forms a group of symmetries for the figure. Every figure, no matter how “nonsymmetric,” has one symmetric group, the trivial symmetry, however, the more symmetric a figure, the larger its symmetry group. A rectangle has four symmetries, whereas the more “symmetrical” square has eight symmetries. Can you find them?

A polygon is called regular if all its sides have the same length and all its interior angles are equal. An equilateral triangle is a regular 3‐gon, a square is a regular 4‐gon, a pentagon a regular 5‐gon, and so on. The symmetry group of a regular n‐gon has n rotational symmetries and n flip symmetries for a total of 2n symmetries. This group is called the dihedral group of the n‐gon and denoted by D_n Can you find the 10 symmetries of the dihedral group D₅ of the pentagon drawn in Figure 6.28?

6.2.7 Multiplying Groups

There are only two (nonisomorphic) groups of order 6, the dihedral group D₃ and the cyclic group ℤ₆. Sometimes groups can be factored into smaller groups somewhat like the number six that can be factored as 6 = 3 × 2. The cyclic group ℤ₆ can be factored as the Cartesian product

of the two smaller cyclic groups ℤ₃ and ℤ₂ where if we define the group operation on the Cartesian product as

we arrive at the Cayley table in Figure 6.29.

Also, if we make the following identification

the multiplication table in Figure 6.29 is the multiplication table for the cyclic group ℤ₆, as shown in Figure 6.30.