411 REFERENCE · Wikibooks
Abstract Algebra/Group Theory/Subgroup/Coset/a Subgroup and its Cosets have Equal Orders
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Theorem
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
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
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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)