Groups. Binary Operations.

There are 11₂ kinds of people: those who understand binary, and those who don’t.

Without pain, without sacrifice, we would have nothing, Fight Club.

Binary Operations

Definition. A binary operation on a set G is a rule, method, or correspondence that assigns to each ordered pair of elements of the set G a uniquely determined element of the set. Therefore, it is a function from the set G x G of ordered pairs (a, b) to G. The value is frequently denoted as a * b, a ∘ b, or simply ab.

Examples:

• Both addition and multiplication are binary operation on ℕ, ℤ, ℤ_n, ℚ, ℝ, and ℂ.
• A division is not a binary operation on the set of natural ℕ, integer ℤ, rational ℚ, real ℝ, and complex numbers ℂ.

  Division is not a binary operation on the set of integers because the result is not always an integer (e.g. 2/3 ∉ ℤ). It is not either a binary operation on the set of real numbers because we cannot divide by zero (1/0 ∉ ℝ).

• Let M(ℝ) be the set of all matrices with real entries. Matrix addition is not a binary operation on M(ℝ) when matrices are incompatible, but it is a binary operation on the set of n x n matrices with real scalars, M_n(ℝ).
• Vector addition is a binary operation on ℝⁿ. Besides, cross product operation on ℝ³ is a binary operation (a x b = ||a|| · ||b|| · sin(θ)), too. However, scalar multiplication of vectors, ℝ x ℝⁿ → ℝⁿ, λ · a = $(λ·a_1, λ·a_2, ···, λ·a_n)$ is not a binary operation because it is the product of a scalar and a vector and it produces a vector. A binary operation requires both the two factors and the result to be on the same set. The dot product of two vectors, ℝⁿ x ℝⁿ → ℝ, a · b = $\sum_{i=1}^n a_ib_i=a_1b_1 + a_2b_2 + ··· + a_nb_n$ is not a binary operation either.
• Let X^X = {f function, f: X → X}, the composition on X^X is a binary operation.

A binary operation is simply a rule, formula or method of combining the elements of a set, two at a time, in such a way that their combination is also a member of the set. This condition is called closure.

For a binary set, a binary operation can be defined or described by a table using the following rule, (ith entry on the left)*(jth entry on the top) = (entry in the ith row and jth column of the table), see Figure 1.c. (e.g., b*c = d, c*d = b).  

Let∗be a binary operation on S, H a subset of S (H ⊂ S). The subset H is closed under∗if the restriction of * to H forms a binary operation on H, that is, ∀a, b ∈ H, we also have a∗b ∈ H.

Examples:

• ℝ is closed under + on ℂ. The same goes as to ℚ (it is closed under addition on ℝ), ℤ is closed under addition on ℚ, and ℕ is closed under addition on ℤ.
• Counterexample: Subtraction is not a binary operation on ℕ, e.g., 2 -3 or 1 -4 are not defined (-1 ∉ ℕ, -3 ∉ ℕ). Futhermore, ℕ ⊂ ℤ is not closed under subtraction. On the other hand, ℤ ⊂ ℚ is closed under subtraction.
• Division is not a binary operation on ℤ or ℚ. ℚ^* = {x ∈ ℚ | x ≠ 0}, ℝ^*, and ℂ^* are closed under multiplication. Division is a binary operation on ℚ^*, ℝ^*, and ℂ^*, but ℤ^* is not closed under division (e.g., 2 / 3 ∉ ℤ^*, 3 / 8 ∉ ℤ^*).
• The set of permutations, S_X ⊆ X^X = {f: X → X}, S_X is closed under composition inside X^X.

Group

A group is an abstraction of the operations of addition and multiplication in arithmetic.💡We want to be able to solve equations such as x + 2 = 5:

1. (x + 2) + (-2) = 5 + (-2). We need inverses.
2. (x + 2) + (-2) = 3. It needs to be closed under addition +.
3. x + (2 + (-2)) = 3. We also need the associative property. Associative property is also known as grouping property.
4. x + 0 = 3. … and the identity.

Definition. A group is a nonempty set G together with a binary operation on G, denoted as ◦, ·, *, or simply omitted, satisfying the following axioms:

1. Associativity. The binary operation is associative; that is, (ab)c = a(bc) for all a, b, c in G. (ab)c = a(bc) ∀a, b, c ∈ G.
2. Identity. There exist an element e (called the identity) in G such that, for every a in G, ae = ea = a. ∃e ∈ G: ∀a ∈ G, ae = ea = a.
3. Inverses. For each element a in G, there exists an element b in G such that ab = ba = e. It is called the inverse of a and is commonly denoted as a^-1. ∀a ∈ G, ∃a^-1∈ G: ab = ba = e.

The definition of a group does not require that ab = ba for every pair of elements a and b. If this additional condition holds, then the binary operation is said to be commutative, and the group is called an Abelian group.

Examples

• ℤ, ℚ, ℝ, and ℂ are all groups under addition. The identity is 0 and -a is the inverse of a. (ℝ^*, ·) is a group, too. The identity is 1 and ∀a ∈ ℝ^*, 1/a is the inverse. However, (ℕ, +) is not a group, 4 and 7 do not have additive inverses (-4 ∉ ℕ, -7 ∉ ℕ).
• (ℚ^*, ·) is also a group. The identity is 1 and ¹⁄_a is the inverse of a. (ℝ^*, ·) and (ℂ^*, ·) are groups, too. However, (ℕ, ·) and (ℤ, ·) are not groups because we do not have inverses (4 and 7 do not have multiplicative inverses, 1/4 ∉ ℕ or 1/7 ∉ ℤ).
• Irrational numbers under multiplication is not a group, e.g., $\sqrt{2}·\sqrt{2} = 2$ ∉ ℝ\ℚ, so it is not closed under multiplication.
• (ℤ, -), (ℚ, -), (ℝ, -), and (ℂ, -) are not groups, e.g, 4 - (3 -7) = 8 ≠ (4 -3) -7 = -6 because subtraction is not associative.
• ℤ/nℤ = ℤ_n, the integers modulo n = {[0], [1], [2], ... [n-1]} is an Abelian group under addition. This is the set of equivalence classes where a ~ b ↭ n | (b - a). Be aware, it is not the set of its representatives.

The identity is [0], the congruence class of all multiples of n. For a congruence class [k], the inverse is the class [-k]. The associative and commutative properties are inherited from ℤ.

• You can multiply elements of Z_n, but you do not obtain a group, e.g., 0 does not have a multiplicative inverse. An integer “a” has a multiplicative inverse modulo n if and only if “a” and n are relative prime. U(n) = ℤ_n^* = {k ∈ ℤ_n | gcd(k, n) = 1} is the group of units in ℤ_n. (U(n), ·) is an Abelian group where the identity is 1.

Example: Figure 1.a. U₁₀ = {[1], [3], [7], [9]}. The reader should notice that gcd(1, 10) = gcd(3, 10) = gcd(7, 10) = gcd(9, 10) = 1. [3]·[7] = [7]·[3] = [1], so [3] = [7]^-1 and [7] = [3]^-1.

The Euclidean algorithm is used to calculate inverses in U(n) because it computes a linear combination ak + bn = 1 ⇒ ak ≡ 1 (mod n), and therefore a = k^-1.

• The dihedral group D_n is the group of symmetries of a regular polygon with n vertices, which includes rotations and reflections or, alternatively, the rigid motions taking a regular n-gon back to itself. D_n = {e, r, r²,…, r^n-1, s, rs, r²s,…, r^n-1s} = ⟨r, s | rⁿ = s² = e, rs = sr^n-1⟩. In particular, the dihedral group D₃ is the symmetry group of the equilateral triangle and D₄ is the symmetry group of the square. 

• Let X be a set, the symmetric group defined over X is the group whose elements are all the bijections from the set X to itself and whose group operation is the composition of functions. S_X = {f: X → X, f is a bijection}

1. Associative: (f · g) · h = f · (g · h)
2. Identity, id: X → X, id(x) = x, ∀x ∈ X.
3. Inverses: ∀f ∈ S_X, ∃f^-1 ∈ S_X: f·f^-1 = f^-1·f = id.

If X is finite, X = {1, 2, 3,…, n}, instead of S_X, the symmetric group of X is denoted as S_n, the symmetric group of degree n. If X is a finite, non-empty set, |S_X| = [By convention] |S_n| = n!

Typically, we use Cauchy’s two-line notation to describe it. One lists all the elements of S in the first row and for each element, its image just below it in the second row. Let f ∈ S_n, $f = (\begin{smallmatrix}1 & 2 & ··· & n\\ f(1) & f(2) & ··· & f(n)\end{smallmatrix})$

Example: f, g ∈ S₆, $f = (\begin{smallmatrix}1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 1 & 4 & 3 & 6 & 5\end{smallmatrix}), g = (\begin{smallmatrix}1 & 2 & 3 & 4 & 5 & 6 \\ 5 & 4 & 3 & 2 & 1 & 6\end{smallmatrix})$

$f · g = (\begin{smallmatrix}1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 6 & 1 & 4 & 2 & 5\end{smallmatrix}) · (\begin{smallmatrix}1 & 2 & 3 & 4 & 5 & 6 \\ 5 & 4 & 3 & 2 & 1 & 6\end{smallmatrix}) = (\begin{smallmatrix}1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 4 & 1 & 6 & 3 & 5\end{smallmatrix})$

• Let V be a vector space over a field F. An automorphism is a bijective homomorphism that maps V to itself. The set of all automorphisms of V is denoted by Aut(V). Aut(V) = {φ:V→V: φ is lineal and bijective} is a group where the group operation is the composition. φ is a mapping that preserves the operations of vector addiction and scalar multiplication, that is, (λφ)(αv + βw) = α(λφ)v + β(λφ)w and (φ + φ’)(αv + βw) = α(φ + φ’)v +β(φ + φ’)w.

• $GL(2, ℝ) = \bigl\{ {{[\bigl(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\bigr)]: a, b, c, d ∈ ℝ, ad - bc ≠ 0}} \bigr\}$ is the group of invertible 2 x 2 matrices with entries in ℝ and non-zero determinant.

The operation is defined as $[\bigl(\begin{smallmatrix}a_1 & b_1\\ c_1 & d_1\end{smallmatrix}\bigr)]·[\bigl(\begin{smallmatrix}a_2 & b_2\\ c_2 & d_2\end{smallmatrix}\bigr)]=[\bigl(\begin{smallmatrix}a_1a_2+b_1c_2 & a_1b_2+b1d_2\\ c_1a_2+d_1c_2 & c_1b_2+d_1d_2\end{smallmatrix}\bigr)]$

∀A, B ∈ GL(2, ℝ), det(A·B) = det(A) * det(B) ≠ 0, so A·B∈ GL(2, ℝ), and therefore, it is closed under matrix multiplication. The identity is $[\bigl(\begin{smallmatrix}1 & 0\\ 0 & 1\end{smallmatrix}\bigr)]$; the inverse of $[\bigl(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\bigr)]$ is $[\bigl(\begin{smallmatrix}\frac{d}{ad-bc} & \frac{-b}{ad-bc}\\ \frac{-c}{ad-bc} & \frac{a}{ad-bc}\end{smallmatrix}\bigr)]$. Therefore, GL(2, ℝ) is a group under matrix multiplication. However, it is non-commutative.

More generally, let F be a field, ∀n ∈ ℤ⁺, let GL_n(F) = GL(n, F) = {A | A is an n x n matrix with entries from F and det(A) ≠ 0}. For any arbitrary n x n matrices A, B, let A·B be the product of matrices as computes by the same rules as when F = ℝ. GL_n(F) is a group under matrix multiplication. It is called the general linear group of degree n, the set of nxn invertible matrices together with the operation of ordinary matrix multiplication.

• ℂ* = {a + bi : a, b ∈ R, a ≠ 0}, is the set of nonzero complex numbers. ℂ^* under the operation of multiplication forms a group.Its identity is 1, and ∀z ∈ ℂ*, $z^{-1}=\frac{a-bi}{a^{2}+b^{2}}$

As you probably should know, real numbers can be represented as points on a one-dimensional number line. Complex numbers, with their real and imaginary components, require a two dimensional complex-number plane. This complex number plane can also be represented with polar coordinates, so z = x + iy = r(cosθ + isenθ). r = |z| = $\sqrt{x^{2}+y^{2}}$, x = r cosθ, and y = rsinθ.

Bibliography

1. NPTEL-NOC IITM, Introduction to Galois Theory.
2. Algebra, Second Edition, by Michael Artin.
3. LibreTexts, Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn (Abstract Algebra), and MathMajor.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
7. Andrew Misseldine: College Algebra and Abstract Algebra.