Lagrange's theorem, in the mathematics of group theory, states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. It is named after Joseph Lagrange. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... Joseph Louis Lagrange (January 25, 1736 – April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia. ...

This can be shown using the concept of left cosets of H in G. The left cosets are the equivalence classes of a certain equivalence relation on G and therefore form a partition of G. If we can show that all cosets of H have the same number of elements, then we are done, since H itself is a coset of H. Now, if aH and bH are two left cosets of H, we can define a map f : aH → bH by setting f(x) = ba^-1x. This map is bijective because its inverse is given by f ^-1(y) = ab^-1y. In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

This proof also shows that the quotient of the orders |G| / |H| is equal to the index [G:H] (the number of left cosets of H in G). If we write this statement as In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...

|G| = [G:H] · |H|,

then, interpreted as a statement about cardinal numbers, it remains true even for infinite groups G and H. In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...

A consequence of the theorem is that the order of any element a of a finite group (i.e. the smallest positive integer k with a^k = e) divides the order of that group, since the order of a is equal to the order of the cyclic subgroup generated by a. If the group has n elements, it follows In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

aⁿ = e.

This can be used to prove Fermat's little theorem and its generalization, Euler's theorem. Fermats little theorem (not to be confused with Fermats last theorem) states that if p is a prime number, then for any integer a, This means that if you start with a number, initialized to 1, and repeatedly multiply, for a total of p multiplications, that number by... In number theory, Eulers theorem (also known as the Fermat-Euler theorem or Eulers totient theorem) states that if n is a positive integer and a is coprime to n, then aφ(n) ≡ 1 (mod n) where φ(n) is Eulers totient function and mod denotes the congruence...

The converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d. The smallest example is the alternating group G = A₄ which has 12 elements but no subgroup of order 6. However, if G is abelian, then there always exists a subgroup of order d. A partial converse for the general case is given by Cauchy's theorem. In mathematics an alternating group is the group of even permutations of a finite set. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... -1...

See also

• Sylow's theorem