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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 G, we have
G2: Identity. There exists an element e of G which has the property
e · x = x · e = x.
G3: Inverse. For every
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
2. Suppose that G is a group and 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).
4. Since the associative law holds, we can omit parenthesis in products. For example we can write
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
Proposition 3: If G is a group, and
Proof: We know that (ab)-1 is the unique element x of G which has the property
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
anam = an+m (m,n Z),(2)
(an)m = anm (m,n Z).
If G is abelian, then(3)
(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
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
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-1 · (a · a) = (a-1 · a) · a
= 1G · a = a = a-1 · a = 1G
Thus, 1G is the only idempotent element of G.
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