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The set of integers. Commutative group. Noncommutative group.

The Concept of a Group

Our main goals in this section are to carefully define the notion of a group, and to explore some elementary logical consequences of our definition.

Definition 1: A group is a nonempty set G together with an associative binary operation · on G which has an identity and such that each element has an inverse with respect to the binary operation. What this means in detail is that the following hold:

G1: Associative Law. For all a,b,c element of G, we have a · (b · c) = (a · b) · c.

G2: Identity. There exists an element e of G which has the property

e · x = x · e = x.

for all x element of G. The element e is called an identity element of G.

G3: Inverse. For every x element of G, the exists y element of G such that

x · y = y · x = e.

The element y is said to be an inverse of x.

Remarks: 1. A group can have only one identity element. For if e and e' are both identity elements of G, then e' = e · e' = e by axiom G2 applied with respect to each of the identity elements e and e'. Therefore we may speak of the identity element of G. We will denote the identity element of G by 1G (or just 1 if G is clear from context). The reader should note the following confusion which could possibly result from our notation. Note that Z is a group with respect to the binary operation of addition. Indeed, we observed earlier that addition is associative. The identity element is 0 since 0 + x = x +0 = x for all x element of Z. Finally, -x is an inverse for x, since x + (-x) = (-x) + x = 0. Thus in this particular example 1G = 0. If we were to use the notation 1 for the identity element in this example, we would be misled into thinking that the identity element is the integer 1, which is not the case. Thus, some care must be exercised in using the notation 1 for the identity element

2. Suppose that G is a group and that x element of G. Then x has only one inverse element. For if y and y' are inverses of x, then we have from the definition of an inverse that

x · y = y · x = 1G,
x · y' = y' · x = 1G.

Therefore, by the first equation,

y' · (x · y) = y' · 1G = y'   (by G2).

On the other hand, by the second equation and the associative law G3,

y' · (x · y) = (y' · x) · y
= 1G · y
       = y    (by G2).

Therefore y = y', and x has only one inverse. Thus, we are entitled to speak of the inverse of x, which we will denote by x-1.

3. 1G-1 = 1G; for by the definition of the identity, we have 1G · 1G = 1G. However, 1G-1 is the only element of G which has the property 1G · 1G-1 = 1G-1 · 1G = 1G.

4. Since the associative law holds, we can omit parenthesis in products. For example we can write a · b · c. A priori, this might mean either a · (b · c) or (a · b) · c. However, these two products are equal by the associative law. Similar remarks hold for longer products such as a · b · c · d. We will go one step further and omit the dot indicating the binary operation when the binary operation is clear from context. Thus, for example, we will write abc instead of a · b · c.

Let us now prove two elementary, but very important, results, which will greatly facilitate doing calculations among elements of a group.

Proposition 2: Let G be a group, and let a,b element of G. Then there exists one and only one x element of G such that ax = b.

Proof: Set x = a-1b. Then ax = aa-1b = 1 · b = b. Thus x, exists. If ax = ax' = b, then a-1ax = a-1ax', so that 1x = 1x', which implies that x = x'. Thus there exists only on x such that ax = b.

Proposition 3: If G is a group, and a,b element of G, then (ab)-1 = b-1a-1.

Proof: We know that (ab)-1 is the unique element x of G which has the property (ab)x = x(ab) = 1. Therefore, it suffices to prove that b-1a-1 has this property. But (ab)(b-1a-1) = a(bb-1)a-1 = a1a-1 = aa-1 = 1. Similarly, (b-1a-1)(ab) = 1. Thus, we see that b-1a-1 has the desired property.

Definition 4: We say that a group G is abelian (or commutative) if ab = ba for all a,b element of G.

If G is an abelian group, we will often denote the group operation by + instead of ·. The typical example of an abelian group is Z with respect to addition. We will meet many examples of groups that are not commutative in the next section.

Definition 5: If a group G contains only a finite number of elements, then we say that G is a finite group. If G is a finite group, then the number of elements in G is called the order of G and is denoted |G|.

It is very convenient to use the following power notation for elements of a group. Let G be a group, and let a element of G, n element of Z. Let us define the power an of a. First, let us define an for n > 0. First, we define a0 = 1G. Next, assuming that an has been defined, let us set an+1 = an · a. By the principle of mathematical deduction, this suffices to define an for all n > 0. Let us define an for n < 0 by an = (a-1)-n. [Note that if n < 0, then -n > 0, so that we have already assigned a meaning to (a-1)-n.] Then we have the usual laws of exponents:

anam = an+m   (m,n element of Z),
(an)m = anm   (m,n element of Z).

If G is abelian, then

(ab)n = anbn.

Note that if G is nonabelian, then (3) need not hold. It is easy to verify (1)-(3) by induction.

Let us make one last remark about our definition of a group. In many books the definition of a group includes an extra axiom, the axiom of closure under multiplication, which asserts that if x and y are in G, then x · y element of G. This axiom is already built into our definition, since we have assumed that multiplication is a binary operation on G. However, in verifying that a given object is a group, one must not forget to verify that multiplication does, indeed, define a binary operation on G. This point is quietly buried in our definition and is easy to gloss over, but is very important in studying subgroups of a group which we will do later. To summarize: In order for G to be a group, the product of any two elements of G must be an element of G.

In the study of groups, there are some other definitions that are convenient to be familiar with:

Definition 6: A semigroup is a nonempty set with a binary operation satisfying closure and associativity. (This is similar to a group definition without requiring the Identity or Inverse.)

An example of a semigroup is the set of all binary strings with the operation of concatenation. This is an important example in Automata Theory. Another example would be the set of positive integers > 1 under addition, however just the positive integers under addition would also be a semigroup. This should not be confused with the concept of a subgroup which is covered in a later chapter.

Definition 7: A monoid is a semigroup with Identity.

An Example of a monoid would be the set of positive integers under multiplication.

Definition 8:: An element a of group G is said to be idempotent if a · a = a.

Proposition 7: A group possesses exactly one idempotent element, the identity element.

Proof: Obviously the identity is idempotent because

1G · 1G = 1G.

Suppose that a is an idempotent element of G, so that a · a = a. Multiplying this equation by the inverse of a we obtain

a-1 · (a · a) = (a-1 · a) · a
= 1G · a = a = a-1 · a = 1G

Thus, 1G is the only idempotent element of G.