# Groups: Defintion of a group

## Understanding the Definition of a Group

• A group is a concept in abstract algebra that involves a set of elements and one operation.
• The set consists of elements, and it should be non-empty.
• The operation is a binary operation meaning it combines two elements to produce another element.
• This binary operation should satisfy certain properties for the set of elements to be a group.

## Properties of a Group

• The concept of closure is one of these properties. For a set to have closure under an operation, the operation of any two elements in the set produces an element that also belongs to the set.
• An operation is associative if for any three elements in the set, the way in which the elements are grouped in operation does not affect the outcome. That is, (a * b) * c = a * (b * c), where a, b, c are elements in the set.
• A set with an operation has an identity element if there’s an element in the set such that when any element in the set is operated with this identity element, the original element remains unchanged. So for any element a in the set, a * e = e * a = a where e is the identity element.
• Every element in the set has an inverse element if for every element a in the set, there exists another element in the set such that when both elements are operated together, the identity element is the result. In other words, for all a in the set, there exists a * a^(-1) = a^(-1) * a = e, where e is the identity element and a^(-1) is the inverse of a.

## Examples of Groups

• An example of a group is the set of integers with the operation of addition. The integers are closed under addition, addition is associative, the identity element is 0, and the inverse of any integer n is -n.
• The set of non-zero real numbers with multiplication as an operation is also a group. Multiplication is associative, the identity element is 1, and the inverse of any real number r is 1/r.
• Note that the set of all real numbers with subtraction as the operation is not a group as subtraction is not associative.

## Group Theory Applications

• Group theory is a vital part of many branches of mathematics and has various applications in fields like physics, chemistry, and computer science.