Lagrange's theorem (group theory)
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This article is about Lagrange's theorem in group theory. See also Lagrange's theorem (number theory).
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup 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 ak = 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.
