The following result is a very important corollary to the fact that the left cosets of a subgroup partition a group.
Lagrange's theorem
|G/H| = |G|/|H|.
Let G be a finite group and H a subgroup of G. Then
In particular, |H| divides |G|.
If H is a subgroup of G, then the quotient |G|/|H| is called the index of H in G.
In Section 6.5 we saw that the order of an element g of G is equal to the order of the subgroup of G generated by g. Thus we find:
Corollary
If G is a finite group and
g 
G, then the order of g
divides |G|.
The following famous
result is a second corollary to Langrange's theorem.
Fermat's little theorem
If p is a prime number, then the multiplicative monoid Z/pZ\{0} is a group. So, for all n not divisible by p,
np-1 = 1 mod p.