Coset

Cosets are either identical or disjoint

Let 𝐺 be a group, 𝑔1,𝑔2 ∈𝐺, and 𝐻 ≤𝐺 be a subgroup. #m/thm/group Then the left cosets 𝑔1𝐻,𝑔2𝐻 are

• identical iff 𝑔−11𝑔2 ∈𝐻
• disjoint otherwise

Proof

Let 𝑔1,𝑔2 ∈𝐺, and 𝐻 ⊆𝐺 be a subgroup. Due to basic Properties,

𝑔−11𝑔2∈𝐻⟺𝑔−11𝑔2𝐻=𝐻⟺𝑔1𝐻=𝑔2𝐻

Next assume there exist ℎ1,ℎ2 ∈𝐻 such that 𝑔1ℎ1 =𝑔2ℎ2, i.e. 𝑔1𝐻 and 𝑔2𝐻 have a common element. Then 𝑔1 =𝑔2ℎ2ℎ−11, whence 𝑔1𝐻 =𝑔2ℎ2ℎ−11𝐻 and since ℎ2ℎ−11 ∈𝐻, it follows 𝑔1𝐻 =𝑔2𝐻 and thus 𝑔−11𝑔2 ∈𝐻. Hence is 𝑔−11𝑔2 ∉𝐻, 𝑔1𝐻 and 𝑔2𝐻 can share no common element.

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