A coset of a subgroup has the same order as the subgroup.

This one is just for practice at constructing a proof. The result is trivial.

Consider a finite subgroup H of G with order n. Then

• H = {h₁, h₂, ..., h_n}

and ∀g &isin G,

• Hg = {h₁g, h₂g, ..., h_ng}

where h₁g, h₂g, ..., h_ng &isin G.

Assume ∃a,b<n, a &ne b such that

• h_ag = h_bg

Then by the cancellation laws,

• h_a = h_b

a contradiction. Therefore, Hg has n distinct elements. That is, the coset has the same order as the subgroup.