I am not sure if this is the right place to ask such a question? But I will give it a try here.
Given two groups $G,H$, there is a problem we encounter very often: whether $H$ is isomorphic to a subgroup of $G$ ? This is generally not an easy question. I have learnt representation theory about one year ago. I am always feeling that it would be somehow helpful to apply representation theory to this problem. But I don't quite know what exactly they are? Are there some known results? Or any thoughts about this?