Complete the proof of Theorem 14.4 by showing that if H is a subgroup of a group G and if left coset multiplication (aH)(bH) = (ab)H is well defined, then Ha ⊆ aH.

Data from Theorem 14.4

Let H be a subgroup of a group G. Then left coset multiplication is well defined by the equation 

(aH)(bH) = (ab)H 

if and only if H is a normal subgroup of G.

Proof:  Suppose first that (aH)(bH) = (ab)H does give a well-defined binary operation on left cosets. Let a ∈ G. We want to show that aH and Ha are the same set. We use the standard technique of showing that each is a subset of the other.

Let x∈ aH. Choosing representatives x ∈ aH and a^-1∈ a^-1 H, we have (xH)(a^-1H) = (xa^-1)H. On the other hand, choosing representatives a ∈ aH and a^-1 ∈ a^-1 H, we see that (aH)(a^-1H) = eH = H. Using our assumption that left coset multiplication by representatives is well defined, we must have xa^-1 = h ∈ H. Then x = ha, so x ∈ Ha and aH ⊆ Ha. 

We tum now to the converse: If His a normal subgroup, then left coset multiplication by representatives is well-defined. Due to our hypothesis, we can simply say cosets, omitting left and right. Suppose we wish to compute (aH)(bH). Choosing a ∈ aH and b ∈ bH, we obtain the coset (ab)H. Choosing different representatives ah₁ ∈ aH and bh₂ ∈ bH, we obtain the coset ah₁bh₂H. We must show that these are the same coset. Now h₁b ∈ Hb = bH, so h₁b = bh₃ for some h₃ ∈ H. 

Thus (ah₁)(bh₂) = a(h₁b)h₂ = a(bh₃)h₂ = (ab)(h₃h₂)