# 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.