# Equivalence relation proof

1. Mar 14, 2012

### tiger4

1. The problem statement, all variables and given/known data
Let H and K be subgroups of the group G. Let a,b \in G and define a relation on G by a ~ b if and only if a = hbk for some h \in H and k \in K. Prove that this is an equivalence relation.

2. Relevant equations
a = hbk

3. The attempt at a solution
The goal is to prove the reflexive, symmetric, and transitive properties of equivalence. I was just hoping someone could help lead me in the right direction of how to start each one. Thanks!

2. Mar 14, 2012

### Oster

Reflexive means a~a. Can you find an element in H and another in K such that a=h.a.k?
Symmetric means a~b => b~a. So if a=h.b.k, you need to show b=h'.a.k' for some h' in H and k' in K.
Just use the definitions....