Lagrange's theorem (group theory)

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Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. Lagrange's theorem is named after Joseph Lagrange.

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.

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

|G| = [G:H] · |H|,

then, interpreted as a statement about cardinal numbers, it remains true even for infinite groups G and H.

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

aⁿ = e.

This can be used to prove Fermat's little theorem and its generalization, Euler's theorem.

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.

[edit] See also

• Sylow's theorem