Group Theory

A group is a set ‘G’ in which two elements are combined to produce the third element of the same set. A set of rational numbers (ℚ), a set of complex numbers (ℂ), and a set of integers (ℤ) are a few examples of groups. 

Group theory involves the study of groups, which can be used to analyze objects or properties that remain invariant under transformation. Algorithms based on group theory also help solve puzzles like the Rubik’s Cube.

Mathematically, a group operation, G × G → G, is represented as (x, y) ↦ x ∗ y that follows the following four conditions or axioms.

Axioms 

Closure

For any x, y Є G, the result of the operation, x ∗ y, is also in G.

Associativity

For any x, y, and z Є G, (x ∗ y) ∗ z = x ∗ (y ∗ z)

Thus, the associativity property disregards the arrangement of parentheses and refers to the product of ‘n’ elements in the group G, a₁ ∗ a₂ ∗ a₃…∗ a_n, since their arrangement does not affect the result. Once the operation is done, the product is written as a₁a₂a₃…a_n. However, the order of the elements matters, as it is always not true that xy = yx for all x,y ∈ G.

Identity

For all a Є G, if there exists an element e Є G such that  e ∗ x = x ∗ e = x holds, then ‘e’ is the identity element of ‘G’

Here, the element ‘e’ in the group ‘G’ is also called the neutral element. If eᐟ Є G  is a second such element, then eᐟ = e ∗ eᐟ = e.

Invertibility

For any x Є G, if there exists an element y Є G such that x ∗ y = y ∗ x = e holds, then ‘y’ is the inverse of ‘x’ in ‘G.’ Here, ‘e’ is the unique element of ‘G’ satisfying x ∗ x = x, which means the element ‘e’ is the inverse of itself.

Examples

A set of rational numbers (ℚ) with multiplication operation, a set of integers (ℤ) with addition operation, and a set of complex numbers (ℂ) with subtraction operations are a few examples of group operations. 

Classes of Groups 

Groups can be classified into various types based on their structure and properties.

Finite Group

A group whose underlying set has a finite number of elements is called a finite group.

Infinite Group

A group whose underlying set contains an infinite number of elements is called an infinite group.

Cyclic Group

A group is said to be cyclic if a single element generates it.

Mathematically, it is expressed as G = 〈g〉for some g Є G.

Abelian Group

A group is said to be abelian or commutative if every two elements commute.

If two elements x and y belong to the group G and the equation x ∗ y = y ∗ x holds, then G is an abelian group.

Others

Group theory may include finite permutation groups, matrix groups, and abstract groups defined by generators and relations. Some more examples of group classes include permutation groups, matrix groups, transformation groups, and abstract groups.

Group Properties 

Order 

The order of a finite group ‘G,’ represented by |G|, is the number of elements in ‘G.’ If ‘k’ is the smallest positive integer such that g^k = e_G, then ‘k’ is said to be the order of the element ‘g’ in G. 

A group is said to be finite if its order is a finite number and infinite if its order is infinite. The finite group whose order is a power of any prime number ‘p’ is called the p-group. However, if no ‘k’ exists, ‘g’ is said to be of infinite order, and ‘G’ is the infinite group.

Homomorphisms

If the function f: G → H is such that for all x and y Є G, f(x ∗ y) = f(x) Δ f(y), it is called the homomorphism of the groups as the function f: G → H has the same domain and codomain.

Isomorphisms 

In abstract algebra, isomorphisms occur when two mathematical objects have the same structure. 

Two groups (G, ∗) and (H, Δ) are said to be isomorphic to each other if there exists a function f: G → H is bijective such that f(x ∗ y) = f(x) Δ f(y) for all x and y Є G. 

Exponentiation 

If g is an element of the group ‘G’ with an inverse g^-1 and k Є ℤ, then

${ g^{m}=\begin{cases}g\ast g\ast \ldots \ast g\left( k \ times\right)  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \ if \ k >0\\ e \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \   \  \\   \  \  \  \  \  \  \  \  \  \  \  \  \  \  \ if \ k=0\\ g^{-1}\ast g^{-1}\ast \ldots \ast g^{-1}\left( k \ times\right)  \  \ if \ k <0\end{cases}}$ 

Here, like the exponential laws of integers, the elements of any finite group hold the usual rules:

g^kg^m = g^{k + m} and (g^k)^m = g^km, for all g Є G, and for all k, m Є ℤ.

Subgroups

A subgroup is a subset of a group but is also a group in itself. 

If (G, ∗) is a group and a non-empty set H ⊆ G such that

• e Є H, where ‘e’ is the identity element of G
• If for all x, y Є H, x ∗ y Є H
• If x Є H, x^-1 Є H

Then, group H is a subgroup of G and H ≤ G.

Lagrange’s group theory connects a group’s order to the order of its subgroups.

Propositions

Here are some propositions that are used to describe groups and their elements.

Proposition 1

If ‘G’ is a group that has the elements ‘x’ and ‘y,’ then (x × y)^-1 = x^-1 × y^-1

Proof

Here, we need to prove (x × y) × y^-1 × x^-1 = e, where ‘e’ is the identity element of the group ‘G’ …..(i)

Now, considering the L.H.S. of the equation (i), we get

(x × y) × y^-1 × x^-1

= x × (y × y^-1) × x^-1 (by the associativity axiom)

= x × e × x^-1 (by the invertibility axiom)

= x × x^-1 (by the identity axiom)

= e (by the invertibility axiom)

Thus, (x × y) × y^-1 × x^-1 = e

⇒ (x × y) × (y^-1 × x^-1) = e …..(ii)

Also, (x × y) × (x × y)^-1 = e …..(iii)

By comparing (ii) and (iii), we get

(x × y)^-1 = y^-1 × x^-1

⇒ (x × y)^-1 = x^-1 × y^-1 (since multiplication holds the commutative property)

Hence, it is proven.

Proposition 2

If ‘x,’ ‘y’, and ‘z’ are three elements in a group ‘G’ such that x × y = z × y, then x = z.

Proof

Let us consider that x × y = z × y …..(i)

Since ‘y’ is an element of the group ‘G,’ there exists another element ‘a’ in ‘G’ such that y × a = e, where ‘e’ is the identity element of the group ‘G’ …..(ii)

On multiplying both sides of (i) by ‘a,’ we get

x × y × a = z × y × a

⇒ x × (y × a) = z × (y × a) (by the associativity axiom)

Using the equation (ii),

a × e = c × e

⇒ a = c (by the identity axiom), proved.

This is also known as cancellation law.

Proposition 3

Let ‘G’ be a group with the subgroups ‘H’ and ‘K’ such that HK = G. If every element of ‘H’ commutes with every element of ‘K’ and H ∩ K = {e}, then G ≅ H × K

Proof

Let us consider a function f: H × K → G such that f: (x, y) ↦ xy

Now, 

f((x₁, y₁), (x₂, y₂)) 

= f(x₁x₂, y₁y₂) 

= x₁x₂y₁y₂ 

= x₁y₁x₂y₂ 

= f(x₁,x₂) f(y₁, y₂)

Thus, the function ‘f’ holds the group operation.

If f(x₁​, y₁) = f(x₂, y₂), then x₁y₁ = x₂y₂​, or x₂^-1x₁ = y₂y₁^-1 …..(i)

We observe that the L.H.S. belongs to ‘H’ by closure, and the R.H.S. belongs to ‘K.’ Since both sides are equal, they must belong to H ∩ K and are equal to the identity element ‘e.’ 

Thus, x₁ = x₂ and y₁ = y₂, which implies ‘f’ is injective.

Since any g Є G is also written as xy for x Є H and y Є K, then ‘f’ is surjective. 

Thus, by the definition, ‘f’ is an isomorphism, which means G ≅ H × K, proved.

• More Resources
  • Lagrange Theorem in Group Theory