# Group Theory: Definition, Examples, Properties

In Group theory, we analyze the algebraic structures of a set with a binary operation given. In this article, we will learn the definition of a group (in Abstract Algebra) with their properties, examples, and applications.

## Definition of a Group

Let G be a set and o be a binary operation acting on it. Then the pair (G, o) is called a group if the following axioms are satisfied.

1. [Closure] G is closed under the composition o. That is, for every a, b ∈ G, we have that aob ∈ G.
2. [Associativity] o is associative. That is, for all a,b,c ∈ G, we have (aob)oc=ao(boc) ∈ G.
3. [Identity] There exists an element e in G such that aoe=eoa=a, for all a ∈ G.
4. [Inverse] For every a ∈ G, there exists a-1 ∈ G such that aoa-1 = a-1oa = e, for all a ∈ G.

In addition to the above four axioms, if (G, o) satisfies the commutative property, that is, aob=boa for all a,b ∈ G, then the group G is called a commutative group or an abelian group (after the name of Norwegian mathematician Niels Henrik Abel).

For example, (Z, +) is an abelian group as we have m+n=n+m for all m,n ∈ Z.

## Examples of Groups

Example 1: (Z, +) is a group, where Z is the set of all integers.

Proof:

Conclusion:

As the above four conditions are satisfied for (Z, +), we can say that the pair (Z, +) is a group. It is called the additive group Z.

More Examples of Groups:

• (R, +) is a group, where R is the set of all real numbers.
• (C, +) is a group, where R is the set of all complex numbers.

## Order of Groups

Let (G, o) be a group. The order of the group G is the cardinality of G, denoted by |G|. If |G| is finite, we say that (G, o) is a finite group. Otherwise, it is called an infinite group.

• (Z, +) is an infinite group as the number of elements of Z is not finite.
• (Z/2Z, +) is a finite group of order 2.

## Types of Groups

There are many types of groups. For example,

• Abelian Group: commutative groups are known as abelian groups
• Cyclic Group: If a group is generated by a single element, then it is called a cyclic group. For example, (Z, +) is a cyclic group generated by 1.
• Prime order groups: If a group has prime order, then it is called a group of prime order. These groups are always cyclic.

## Properties of Groups

Every group (G, o) satisfies the following properties:

1. The composition of two elements always belongs to G. That is, aob∈G for all a,b in G.
2. Every group (G, 0) contains only one identity element.
3. Each element in a group (G, o) has only one inverse.
4. In a group (G, o), the cancellation law holds.
• aob=aoc ⇒b=c (left cancellation law)
• boa=coa ⇒b=c (right cancellation law)
5. We have (aob)-1 = b-1oa-1 for all a,b ∈G. That is, the inverse of ab is equal to b-1a-1.

## Applications of Group Theory

Group theory has many applications in Physics, Chemistry, Mathematics, and many other areas. It plays a crucial role in public-key cryptography.

## FAQs

Q1: What is the inverse of the identity in a group?

Answer: The inverse of identity is the identity itself. That is, e-1=e.

Q2: Is (2Z, +) a group?

Answer: Yes, (2Z, +) is a commutative group.

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