Question

Let H be a subgroup of G. For a, b ∈ G, let a ∼ b if and only if ab^-1 ∈ H. Show that ∼ is an equivalence relation on G.​

Answer 1

Answer:

We recall from Section 11, the equivalence relation congruence modulo n on

the set of integers. So if a ≡ b(mod n) then n|(a − b) which means that

a − b = nq for some q ∈ Z. Since < n >= {nt : t ∈ Z} then it follows that

a − b ∈< n >, where < n > is a subgroup of Z. Thus, we have

a ≡ b(mod n) ⇐⇒ a − b ∈< n > .

Now −b is the inverse of b in the additive group Z; so by replacing Z by

a group G and < n > by a subgroup H of G the above relation suggests

considering the relation defined by

a ∼ b ⇐⇒ ab−1 ∈ H

Theorem 16.1

If H is a subgroup of a group G, the relation ∼ defined above is an equivalence

relation on G. The equivalence class with representative a is the set