TheoremsDefinitionsArticles

Understanding Groups

A group is one of the fundamental structures in abstract algebra. The concept allows us to study symmetry in a mathematical way.

For example, the integers mathbbZ\\mathbb{Z} under addition form a group. We can verify each axiom:

  • Closure: The sum of any two integers is an integer
  • Associativity: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)
  • Identity: 00 is the additive identity
  • Inverse: Every integer aa has an inverse −a-a
mathbbZ=ldots,−2,−1,0,1,2,ldots\\mathbb{Z} = \\{\\ldots, -2, -1, 0, 1, 2, \\ldots\\}

You can reference Theorem 1 and Definition 1 from the other tabs to understand properties of groups.