In an experiment what should be the relationship between the control group and experimental group?

Ideally, the experimental and control groups are identical. As much as possible, the two groups should overlap in every characteristic.

Explanation:

At the beginning of your study, you need your two groups to be as statistically similar as is humanly possible. That way, when you control for other variables, you can reasonably attribute any statistically significant differences at the end of the study to your intervention.