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:

1.	We know that for any two integers a and b, we always have a+b∈Z. Thus, the set Z is closed under the operation +.
2.	For any three integers a, b, and c, we have (a+b)+c=a+(b+c). So + is associative on Z.
3.	We have a+0=a for all a∈Z. Thus, 0 is the identity element in Z. In other words, the identity element exists in Z.
4.	For any a∈Z, a+(-a)=0. So -a is the inverse of a in Z. That is, the inverse exists for every element in Z.

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.

Non-Examples of Groups

1	Z is not a group under multiplication, that is, (Z, ×) is not a group. This is because 2 does not have an inverse in Z as 1/2 ∉ Z.
2	The set of real numbers R is not a group under multiplication, that is, (R, ×) is not a group. This is because 0 does not have an inverse in R. Note that (R-{0}, ×) is a group with the identity 1.

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.

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