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.