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.
Proof of Lagrange's Theorem
This can be shown using the concept of left cosets
. The left cosets are the equivalence classes
of a certain equivalence relation
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
are two left cosets of H
, we can define a map f
by setting f
) = ba-1x
. This map is bijective
because its inverse is given by f −1
) = 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.
Using the theorem
A consequence of the theorem is that the order of any element a
of a finite group (i.e. the smallest positive integer k
) divides the order of that group, since the order of a
is equal to the order of the cyclic
. 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. These special cases were known long before the general theorem was proved.
The theorem also shows a group of prime order is cyclic and simple.
Existence of subgroups of a certain order
Lagrange's theorem raises the question of whether every divisor of the order of a group is the order of a subgroup. This need not hold. Given a finite group G
and a divisor d
|, there does not necessarily exist a subgroup of G
with order d
. The smallest example is the alternating group G
which has 12 elements but no subgroup of order 6. Any finite group which has a subgroup with
size equal to any (positive) divisor of the size of the group must be solvable
, so nonsolvable groups are examples of this phenonenon, although A4
shows that they aren't the only examples.
, then there always exists a subgroup of any order dividing the size of G
. A partial generalization is given by Cauchy's theorem
Lagrange did not prove Lagrange's theorem in its general form.
What he actually proved was that if a polynomial in n
variables has its variables permuted in all n
! ways, the number of
different polynomials that are obtained is always a factor of n
!. (For example if the variables x
, and z
are permuted in all 6 possible ways in the
then we get a total of 3 different polynomials: x
, and y
. Note 3 is a factor of 6.)
The number of such polynomials is the index in the symmetric group Sn
of the subgroup H
of permutations which
preserve the polynomial. (For the example of x
, the subgroup H
contains the identity and the transposition (xy
So the size of H
!. With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to
the general theorem about finite groups which now bears his name.