Groups
In this section, we introduce a different kind of algebraic structure called a group, that uses a single operation rather than two as in rings. Evariste Galois (1811–1832) was the first to develop the concept of a group in 1831, in connection with his reasearch regarding the solutions of polynomial equation. The first formal definition of group will appear only in 1893, in a publishing of Heinrich Weber.
7.1.1 Definition. A group (G, *) is a nonempty set G equipped with a binary operation *,
G x G ⟶ G
(a,b) ⟼ a * b
that satisfies the following axioms.
* is associative, i.e.
(a * b) * c = a * (b * c) ∀ a,b,c ∈ G;
There exists an element e ∈ G (called the identity element) such that a * e = a = e * a for every a ∈ G.
For each a ∈ G, there is an element d ∈ G (called the inverse of a) such that a * d = e and d * a = e.
Commutativity: a * b = b * a, ∀a,b ∈ G.
A group is said to be abelian if it also satisfies this axiom:
Abelian groups are named after the Norwegian mathematician Niels Henrik Abel (1802–1829) who, in spite of his short life, made fundamental contributions to mathematics.
Functional notation p = f(a,b) isn't very convenient for laws of composition. Instead, the element obtained by applying the law to a pair (a,b) is usually denoted using a notation resembling those used for multiplication or addition:
p = ab, a x b, a ∘ b, a ⋅ b, a + b, and so on
We call the element p the product or sum of a and b, depending on the notation chosen.
Examples 7.1.2.
The set (ℤ, +), is a group. Indeed
∀a, b, c ∈ ℤ, a + (b + c) = (a + b) + c, (+ is asociative).
∀a ∈ ℤ, a + 0 = 0 + a = a, (0 is the identity).
∀a ∈ ℤ, a + (−a) = −a + a = 0, (−a is the inverse of a).
Moreover the group is abelian:
∀a, b ∈ ℤ, a + b = b + a (+ is commutative).
(ℤ, ⋅), is not a group with respect to multiplication, even if the associative property holds true and there exists the identity 1, each element of the group has not inverse.
Also the following rings are abelian groups: (ℚ,+), (ℝ,+), (ℂ,+), (ℤ[i],+), (ℤ[√n],+), (ℤ[n,+), (R[x],+), with R commutative ring with identity.
(U(ℤn), ⋅): The set Un of invertible elements of ℤn is a group. Suppose [a] and [b] are elements of U(ℤn) and that their inverses are [a'] and [b'], respectively. Then [ab] is invertible and its inverse is [a'b']. To check this, it suffices to multiply the two elements:
[ab] ⋅ [a'b'] = [a'a] ⋅ [bb'] = [1]
It remains to verify that this operation satisfies the group axioms. Associativity is easy, because we already know that multiplicaton in ℤn is associative. The identity elements is [1], which is an invertible element of ℤn thus belongs to U(ℤn). So the set U(ℤn) under the operation of multiplication is indeed a group. Since ℤn is a set of n elements, U(ℤn) is of finite order and clearly abelian. ■
Other examples of groups are:
Nonnull elements of a field with respect to multiplication: (ℚ\{0},⋅), (ℝ\{0},⋅), (ℂ\{0},⋅), (ℤp,⋅), (p prime)
The set (Mmn(R), +) := {of matrices m x n on a ring R}.
The set (Mn(R), +) := {square matrices n x n on a ring R}.
(GLn(ℝ), ⋅) = {A ∈ Mn(ℝ) | det A ≠ 0} = real linear general group. In linear algebra this is the group of all invertible matrices n × n, with real values, with positive n.
(SLn(ℝ), ⋅) = {A ∈ Mn(ℝ) | det A = 1} = real linear special group.
(On(ℝ), ⋅) = {A ∈ Mn(ℝ) | det AT = A−1} = orthogonal group.
(SOn(ℝ), ⋅) = {A ∈ On(ℝ)| det A = 1}
The first two groups are abelian, whilst the others are not for n ≥ 2.
7.1.2 Proposition. Let (G, ⋅) a group. Then the identity element e ∈ G, is unique.
Proof. Suppose that e and u both serve as identities in G. Then
eg = ge = g ∀g ∈ G
ug = gu = g ∀g ∈ G
The relations
eu = ue = e
ug = gu = u
imply u = e.□
Proposition 7.1.3. Let (G, ⋅) a group. Then the inverse of every element a ∈ G, is unique.
Proof. Suppose that a' and a'' are both inverses of a ∈ G. Then
a' = ea' = (a''a)a' = a''(aa') = a''e = a''. □
Corollary 7.1.4. Cancellation holds in G.
If ab = ac, then b = c; if ba = ca, then b = c.
Proof. By definition of a group, the element a has at least one inverse a' such that a'a = e = aa'. If ab = ac, then a'(ab) = a'(ac). By associativity and the properties of inverses and identities,
(a'a)b = (a'a)c
eb = ec
b = c. □
Corollary 7.1.5. For each a,b ∈ G, the inverse of the product of two elements of G is the product of their inverses in reverse order, that is
(ab)−1 = b−1a−1
Proof. We have to show that the product of b−1a−1 and ab (in both orders) is equal to the identity e of the group. We have
(ab) (b−1a−1) = a−1 ⋅ (b−1b) ⋅ a−1 = a−1 ⋅ e ⋅ a−1 = a−1 ⋅ a = e.
and
(b−1a−1)(ab) = b−1 ⋅ (a−1a) ⋅ b = b−1 ⋅ e ⋅ b = b−1 ⋅ b = e.
Hence by definition (ab)−1 = b−1a−1. □
Corollary 7.1.6 A group G is Abelian if and only if (ab)−1 = a−1b−1 ∀a,b ∈ G.
Proof. If we assume (ab)−1 = a−1 b−1 then we have owing to the previous corollary
(ab)−1 = b−1a−1 = a−1b−1
and so we can argue as follows:
b−1a−1 = a−1b−1
b · b−1a−1 = b · a−1b−1
a−1 = ba−1b−1
a · a−1 = a · ba−1b−1
e = aba−1b−1
e · b = aba−1b−1 · b
b = aba−1
b · a = aba−1 · a
ba = ab
Hence G is Abelian. □