Abstract Algebra/Group Theory/Subgroup/Coset/a Subgroup and its Cosets have Equal Orders
< Abstract Algebra  Group Theory  Subgroup
Theorem[edit]
Let g be any element of group G.
Let H be a subgroup of G. Let o(H) be order of group H.
Let gH be coset of H by g. Let o(gH) be order of gH
 o(H) = o(gH)
Proof[edit]
Overview: A bijection between H and gH would show their orders are equal.
 0. Define
f is surjective[edit]
 1. f is surjective by definition of gH and f.
f is injective[edit]

2. Choose such that  3.
0.  4.
, and subgroup  5.
3. and cancelation justified by 4 on G
o(H) = o(gH)[edit]
As f is surjective and injective,

 6. f is a bijection from H to gH

 7. Such bijection shows o(H) = o(gH)