Cosets are either identical or disjoint

In the same way that equivalence classes are either identical or disjoint, so are cosets.

Consider a subgroup H of G, and elements a, b &isin G. Then H has right cosets Ha and Hb.

Either:

• Ha &cap Hb = ∅, in which case there's nothing more to show

OR:

• ∃c &isin G s.t. c ∈ Ha &cap Hb

In the second case,

• c = h₁a = h₂b for some h₁, h₂ &isin H

and so

• a = h₁^-1h₂b = h₃b, where h₃ = h₁^-1h₂ &isin H

and

• b = h₂^-1h₁a = h₄a, where h₄ = h₂^-1h₁ &isin H