Lagrange's Theorem
From AoPSWiki
Lagrange's theorem is a result on the indices of cosets of a group.
Theorem. Let 
be a group, 
a subgroup of , and 
a subgroup of . Then
Proof. For any 
, note that 
; thus each left coset mod is a subset of a left coset mod ; since each element of is in some left coset mod , it follows that the left cosets mod are unions of left cosets mod . Furthermore, the mapping 
induces a bijection from the left cosets mod contained in an arbitrary -coset 
to those contained in an arbitrary -coset 
. Thus each -coset is a union of -cosets, and the cardinality of the set of -cosets contained in an -coset is independent of the choice of the -coset. The theorem then follows. 
By letting be the trivial subgroup, we have
In particular, if is a finite group of order and is a subgroup of of order 
,
so the index and order of are divisors of 
.
