# The Frobenius-Nakayama Formula

I am currently reading a paper where it refers to the usual Frobenius-Nakayama formula describing quotients of an induced module. It is refering to the following result:

If $k$ is a field, $P$ is a subgroup of a group $G$, $F$ is a $kP$-module and $W$ is a $kG$-module, then we have that $\operatorname{Hom}_{kP}(F,W_{P})\cong_{k}\operatorname{Hom}_{kG}(\operatorname{Ind}_{P}^{G}(F),W).$

However, having searched for the Frobenius-Nakayama formula, I cannot find it in any reliable sources. Does anybody know of a good source for this result, and/or if the result is normally referred to by an alternative name?

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Another name for this result (or rather, something that implies it) is "induction is left-adjoint to restriction." –  mt_ Jun 11 '12 at 14:33

Over a field of characteristic not dividing the group order, this is called Frobenius reciprocity. The version you give is usually called Frobenius-Nakayama reciprocity.

See page 58-59 of Alperin's Local Representation Theory for a proof.

A very introductory treatment is given on page 231 of James–Liebeck's Representations and Characters of Groups.

A more general version is proved on page 46 of Benson's Representations and Cohomology.

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Thanks for a very quick and helpful response. The reference to Benson's book is extremely useful. –  David Ward Jun 11 '12 at 14:02
No prob. I added two more treatments. A should be better if you are doing group theory. J-L should be better if you're just learning character theory, and B might be better if you are doing representation theory. –  Jack Schmidt Jun 11 '12 at 14:04