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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?

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if generally not applicable, I would restrict my question in finite groups. Thanks –  Easy Jan 29 '13 at 11:00
    
I can imagine that you might be lucky and get a negative answer quickly if you knew the character tables of $G$ and $H$, and could easily check that there was no way that the restriction of some irreducible character of $G$ could be a sum of characters of $H$. But I don't see how representation theory could enable you to prove that $H$ really is a subgroup of $G$, –  Derek Holt Jan 29 '13 at 14:36
    
In general I don't think it would work as the character table doesn't contain full information of a group. But is there any sufficient condition which implies $H<G$? For example, if we know $H$ is generated by not many elements (maybe say $H=\langle a,b\rangle$), and assuming $G$ is just simply a general linear group, what can we say about $H$ and $G$? –  Easy Jan 30 '13 at 3:58

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