# Abstract groups

## Defining Abstract Groups

• A group is a set of elements combined with a binary operation that satisfies four fundamental properties: closure, associativity, identity and inversibility.
• The closure property states that for any two elements in the group, their product (result of the operation) is also in the group.
• Associativity requires that for any three elements in the group, (a • b) • c = a • (b • c) where • represents the operation.
• The identity property states that there exists an element, often denoted as ‘e’, in the group, such that for every element ‘a’ in the group, a • e = e • a = a.
• The inverse property means that for every element in the group, there exists another element that, when combined with the first under the group operation, yields the identity element.

## Properties of Abstract Groups

• Groups can be finite, having a limited number of elements, or infinite. The order of a group is the total number of its elements.
• Any subset of a group that is itself a group under the same operation is called a subgroup.
• A cyclic group is a group that is generated by a single element. That element and its powers under the operation form all the elements in the group.
• The symmetric group on a set is the group consisting of all the possible permutations of the set.
• An Abelian (or commutative) group is a group in which the binary operation is commutative, i.e., a • b = b • a for all a, b in the group.

## Examples of Abstract Groups

• Integers under addition form a group, with zero as the identity element and the negative of each integer as its inverse.
• Non-zero real numbers under multiplication form a group with identity as ‘1’ and each element’s inverse being its reciprocal.
• The set of all 2x2 invertible matrices forms a group under multiplication.
• The set {0, 180°} under addition modulo 360° forms a group with 0° as the identity and each angle being its own inverse.
• Symmetric groups: For a finite set of ‘n’ elements, the set of all possible permutations forms a group under function composition, known as the symmetric group of order ‘n!’, denoted as Sn.