Abstract Algebra/Group Theory/Subgroup/Coset/a Subgroup and its Cosets have Equal Orders

TheoremEdit

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)

ProofEdit

Overview: A bijection between H and gH would show their orders are equal.

0. Define  

f is surjectiveEdit

1. f is surjective by definition of gH and f.

f is injectiveEdit

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)