Abstract Algebra/Group Theory/Subgroup/Coset/a Subgroup and its Cosets have Equal Orders
Jump to navigation Jump to search
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)
Overview: A bijection between H and gH would show their orders are equal.
- 0. Define
f is surjective
- 1. f is surjective by definition of gH and f.
f is injective
o(H) = o(gH)
As f is surjective and injective,
- 6. f is a bijection from H to gH
- 7. Such bijection shows o(H) = o(gH)