What defines a group

Understanding Groups

A group in mathematics is a set combined with an operation that satisfies certain conditions or rules.
These rules make the concept of groups vital in different areas of mathematics such as algebra, geometry and number theory.

Properties of a Group

A set forms a group under a specific operation if it fulfils four axioms or properties: closure, associativity, the existence of an identity element, and the existence of inverse elements.

The Closure Property

Closure under an operation means that for any two elements in the set, the result of the operation is also a member of the set.
For example, in the group of integers under addition, the sum of any two integers is always another integer.

Associativity of Operation

The associativity property indicates that for any three elements in the set, the way they are grouped in an operation does not affect the result.
In the group of real numbers under multiplication, for instance, (ab)c is the same as a(bc).

Existence of Identity Element

A group must contain an identity element, or a element that leaves any other element unchanged when the operation is applied between them.
In the group of integers under addition, the identity element is zero as adding zero to any integer doesn’t change its value.

Existence of Inverse Elements

Every element in a group must have an inverse inside the group such that the operation between an element and its inverse results in the identity element.
For example, in the group of integers under addition, the additive inverse of any integer ‘a’ is ‘-a’ because ‘a’ added to ‘-a’ equals zero.

Working with Groups

Understanding how to identify and use groups will be helpful for tackling different mathematical problems, especially in algebra.
Practice working with group structures in varied problems to strengthen comprehension.

Advanced Groups

Some groups also fulfil the commutative property and are called Abelian groups.
Commutativity means that the order in which elements are processed in the operation doesn’t affect the result.
For instance, the group of integers under addition is also an Abelian group as ‘a + b’ always equals to ‘b + a’ for any integers ‘a’ and ‘b’.

It is recommended to use various resources, delve into numerous examples and work on past papers to enhance the understanding of groups.