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Groups

In mathematics, a group is a set provided with an operation that connects any two elements to compose a third element in such a way that the operation is associative, an identity element will be defined, and every element has its inverse. These three conditions are group axioms, hold for number systems and many other mathematical structures. For instance, the set of integers and the addition operation form a group. That means, in simple words, a group is a combination of a set and binary operation.

Before learning in detail about groups, let’s understand the basics that are required to define groups in mathematics.

Set: A set is a collection of well-defined things, objects or elements, and it does not vary from person to person and is represented using a capital letter.

Binary operation: An operation that contains two elements of a set given another element of the same set is called a binary composition or binary operation. Generally, a binary operation can be represented using “.” or “⋆”.

Click here to learn more about the binary operation.

Algebraic structure:

An algebraic structure is a set of elements, i.e., the carrier of the structure with an operation that matches any two members of the set uniquely onto a third member. One of the most basic algebraic structures is the group. The axioms give the specificity of an algebraic structure that it satisfies.

Let’s have a look at the mathematics definition of groups.

Groups Definition

If G is a non-empty set and “⋆” is the binary operation defined on G such that the following laws or axioms are satisfied then, (G, ⋆) is called a group.

Let “G” be a non-empty set and “⋆” be a binary operation on G such that

(G1) – Closure law

for a, b ∈ G, a ⋆ b ∈ G

(G2) – Associative law

a ⋆ (b ⋆ c) = (a ⋆ b) ⋆ c for all a, b, c ∈ G

(G3) – Identity element

there is an element e ∈ G such that a ⋆ e = e ⋆ a = a for all a ∈ G; where e is the identity element

(G4) – Inverse law

for each a ∈ G, there exists an element b ∈ G such that a ⋆ b = b ⋆ a = e, where b = a-1 is the inverse element of a.

Notation and Examples of Groups

Consider a set of real numbers and a binary operation, say addition, then it forms a group. This can be represented as (ℝ, +). Similarly, (Z, +) is also a group that comprises a set of integers under addition.

Terminology of Groups

We can define various terms related to groups based on the number of laws they satisfy.

Abelian Group or Commutative Group

(G; ⋆) is said to be an abelian group, or a commutative group is a binary operation that satisfies the commutative law, i.e., a ⋆ b = b ⋆ a for all a, b ∈ G.

Semi Group

If the set G satisfies only closure law and associative law, then G is called a semi-closed group or semi group.

Finite and Infinite Group

In a group, G contains only a finite number of elements, then group G is called a finite group; otherwise, group G is called an infinite group.

Order of a group: The number of elements in a finite group G is called the order of a group and is denoted by O(G). That means if the number of elements in G is n, then O(G) = n.

Theorem on Groups

Theorem 1:

In a group, the identity element is unique (or) Uniqueness of the identity element.

Proof:

Let (G, ⋆) be a group.

Let us assume that e and f are the two identity elements of group G.

Case (i):

Let e ∈ G be the general identity element of group G.

And f ∈ G be the identity element of group G.

⇒ e ⋆ f = f ⋆ e = e….(1)

Case (ii)

Let f ∈ G be the general identity element of group G.

And e ∈ G be the identity element of group G.

⇒ f ⋆ e = e ⋆ f = f….(2)

From (1) and (2),

e = f

Hence, in group G, the identity element is unique.

Theorem 2: (Cancellation laws)

Let G be a group, then for a, b, c ∈ G;

ab = ac ⇒ b = c (Left cancellation law)

ba = ca ⇒ b = c (Right cancellation law)

Proof:

Let G be a group and e be the identity element in group G.

Also, consider aa-1 = a-1a = e….(1)

Left cancellation law:

Consider ab = ac

⇒ a-1(ab) = a-1(ac)

⇒ (a-1a)b = (a-1a)c

⇒ eb = ec [From equation (1)]

⇒ b = c

Thus, ab = ac ⇒ b = c

Hence, the Left cancellation law is proved.

Right cancellation law:

Consider ba = ca

⇒ (ba)a-1 = (ca)a-1

⇒ b(aa-1) = c(aa-1)

⇒ be = ce [From equation (1)]

⇒ b = c

Thus, ba = ca ⇒ b = c

Hence, the Right cancellation law is proved.

Frequently Asked Questions on Groups

What is a group in maths?

In maths, a group is the combination of a set and binary operation. For example, the set of integers with an addition operation forms a group and a set of real numbers with a binary operation; addition is also a group. These satisfy some laws, say closure, associative, identity and inverse to represent as a group.

Is (Z, -) a group?

(Z,−) denotes the algebraic structure initiated by the set of integers under the subtraction operation. However, (Z,−) is not a group since it does not satisfy the commutative law. For example, 5 – (-1) is not equal to (-1) – 5.

What is the order of a group?

The number of elements in a finite group G is called the order of a group. It is denoted by O(G).

Is (N, +) a group?

(N,+) denotes the algebraic structure initiated by natural numbers under the addition operation. However, (N,+) is not a group since it does not satisfy the inverse law. For instance, 3 ∈ N has no inverse element, i.e., -3 in N with respect to +.

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