Clearly, hk = kh iff hkh^{-1}k^{-1} = e. Prove this by showing that hkh^{-1}k^{-1} ∈ H ∩ K.
I don't know How?!. Can you tell me How?
I attached my solution I think it is not correct but you might be able to see what I did and tell me how to correct it
CCI04202013_00003.bmp
I don't think it works. First, you did not properly introduce a and b: what exactly are they? More seriously, you started with hk = kh and arrived at h_{2}k_{2} = k_{2}h_{2}. You assumed what you are supposed to prove.
The idea is to note that hk = kh ⇔ hkh^{-1}k^{-1} = e ⇔ (hkh^{-1}k^{-1} ∈ H and hkh^{-1}k^{-1} ∈ K). This has been described several times in this thread.