Group (mathematics)

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A group is a mathematical structure consisting of a set of elements combined with a binary operator which satisfies four conditions:

  1. Closure: applying the binary operator to any two elements of the group produces a result which itself belongs to the group
  2. Associativity: (AB)C = A(BC) where A, B and C are any element of the group
  3. Existence of Identity: there must exist an identity element I such that IA = AI = A; that is, applying the binary operator to some element A and the identity element I leaves A unchanged
  4. Existence of Inverse: for each element A, there must exist an inverse A − 1 such that AA − 1 = A − 1A = I

A group with commutative binary operator is known as Abelian.

Examples

  1. the set of integers \mathbb{Z} under addition, (\mathbb{Z},+): here, zero is the identity, and the inverse of an element a \in \mathbb{Z} is − a.
  2. the set of the positive rational numbers \mathbb{Q}_+ under multiplication, (\mathbb{Q}_+,\cdot): 1 is the identity, while the inverse of an element \frac{m}{n} \in \mathbb{Q}_+ is \frac{n}{m}.
  3. for every n \in \mathbb{N} there exists at least one group with n elements,e.g., (\mathbb{Z}/n\mathbb{Z},+) = (\mathbb{Z}_n,+).
  4. the set of complex numbers {1, -1, i,-i} under multiplication, where i is the principal square root of -1, the basis of the imaginary numbers. This group is isomorphic to \mathbb{Z}_{4} under mod addition.
  5. the Klein four group consists of the set of formal symbols {1,i,j,k} with the relations i^{2} =j^{2}=k^{2}=1, \; ij=k, \; jk=i, \; ki=j. All elements of the Klein four group (except the identity 1) have order 2. The Klein four group is isomorphic to \mathbb{Z}_{2} \times \mathbb{Z}_{2} under mod addition.
  6. the set of "moves" on a Rubik's cube, where a move is understood to be a finite sequence of twists: here, the identity move is to do nothing, while the inverse of a move is to do the move in reverse, thereby undoing it.

Groups are the appropriate mathematical structures for any application involving symmetry.

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