Groups: Lagrange's theorem

Groups: Lagrange’s theorem

Understanding Lagrange’s Theorem

  • Lagrange’s theorem is a fundamental theorem in group theory, a branch of abstract algebra.
  • It states: if ‘H’ is a subgroup of a finite group ‘G’, the order (number of elements) of ‘H’ divides the order of ‘G’.
  • In other words, the size of any subgroup of a finite group must be a divisor of the size of the original group.
  • The order of a group or a subgroup is the number of elements it contains.
  • A subgroup H of a group G is a subset of G that is itself a group under the operation of G.

Lagrange’s Theorem and Group Order

  • A consequence of Lagrange’s theorem is that the order of any element of a finite group (i.e., the smallest positive integer n such that ‘a^n=e’, where ‘a’ is a group element and ‘e’ is the identity) divides the order of the group.
  • Lagrange’s theorem doesn’t guarantee that for each divisor of the order of a group, there is a subgroup of that order. Hence, care should be taken while using this result.

Applying Lagrange’s Theorem

  • To apply Lagrange’s theorem, firstly identify the group and its subgroup.
  • Calculate the order of both the group and the subgroup.
  • Check if the order of the subgroup divides the order of the group. Its validity confirms the given set is a subgroup.

Lagrange’s Theorem in Abstract Algebra

  • Abstract algebra is a branch of mathematics which studies algebraic structures and their properties.
  • Lagrange’s theorem is one of the most important theorems in abstract algebra, and is a classical result in finite group theory.
  • Lagrange’s theorem provides the basis for many proofs and subsequent theorems, including the Fundamental Theorem of Finite Abelian Groups and subsequent properties such as group actions and group isomorphisms.

Cadlag Functions and Lagrange Polynomials

  • Note that ‘Lagrange’ refers to Joseph-Louis Lagrange, hence, in mathematics, we also have other references to his work such as the Lagrange polynomials in numerical analysis or Lagrange multipliers in optimisation.
  • Do not confuse Lagrange’s Theorem in group theory with these topics, as they belong to different areas of study within mathematics.