Linear Algebra Problem?
Let G be a group and H a subgroup.
Prove that the relation a≅b if and only if ab^(-1) ∈ H is an equivalence relation.
- No MythologyLv 77 years agoFavorite Answer
It's critical that H is a subgroup making it closed under the group operation and inversion.
(a) H must contain the identity. Note that aa^(-1) = e ∈ H. So a ≅ a for each a in G.
(b) Note that (ab^(-1))^(-1) = (b^(-1))^(-1)a^(-1) = ba^(-1). This is a general property of group inverses, but it's key here to showing symmetry.
Suppose a ≅ b so that ab^(-1)∈ H. Since H is closed under inversion, ba^(-1) = (ab^(-1))^(-1) ∈ H. Thus we have a ≅ b implies b ≅ a.
(c) Now use the fact that H is closed under the group operation. Suppose a ≅ b and b ≅ c for some a, b, and c in G. Then
ab^(-1) ∈ H and bc^(-1) ∈ H.
Since H is closed, the product of these two elements is in H. That is
(ab^(-1))(bc^(-1)) = ab^(-1)bc^(-1) = aec^(-1) = ac^(-1) ∈ H.
(Used the associativity property there in the middle.) Thus a ≅ c whenever a ≅ b and b ≅ c.
We've established (a) reflexivity, (b) symmetry, and (c) transitivity.