The most general form of Lagrange's group theorem, also known as Lagrange's lemma, states that for a group 
, a subgroup 
of 
, and a subgroup 
of 
, 
, where the products are taken as cardinalities
(thus the theorem holds even for infinite groups)
and 
denotes the subgroup index for the subgroup 
of 
. A frequently stated corollary (which follows from taking
,
where 
is the identity element) is that the order of
is equal to the product of the order of 
and the subgroup index of
.
The corollary is easily proven in the case of 
being a finite group, in which
case the left cosets of 
form a partition of 
, thus giving the order of 
as the number of blocks in the partition (which is 
) multiplied by the number of elements in each partition
(which is just the order of 
).
For a finite group 
, this corollary gives that the order of 
must divide the order of 
. Then, because the order of an element 
of 
is the order of the cyclic subgroup generated by 
, we must have that the order of any element of 
divides the order of 
.
The converse of Lagrange's theorem is not, in general, true (Gallian 1993, 1994).
