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I'm learning what irreducible representation is and need some examples. One of them is as follows:
Let $G$ be any group and $H$ - it abelian subgroup. How to prove that each irreducible representation of $G$ is a subrepresentation of someone, induced from one-dimensional representation of $H$ (this is a problem in the problem sheet)?

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Hint: All irreducible representations of $H$ are one-dimensional. Then use Frobenius reciprocity. – Tobias Kildetoft Dec 21 '13 at 12:23
    
Could you give some details briefly? – user74574 Dec 21 '13 at 12:59
    
Are you familiar with Frobenius reciprocity? – Tobias Kildetoft Dec 21 '13 at 13:00
    
I saw the formula:) – user74574 Dec 21 '13 at 13:01
    
That formula has as a special case that if $\psi$ is a constituent of $\chi_H$ then $\chi$ is a constituent of $\psi^G$ (where $_H$ means restriction and $^G$ means induction). – Tobias Kildetoft Dec 21 '13 at 13:03

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