# Is relatively free the same thing as induced for finite group modules?

I was looking over Alperin's Local Representation Theory and I realized I remembered a definition that may not be there (or true).

Is a relatively H-free G-module exactly the same as a G-module isomorphic to an induced H-module?

Lemma 8.4 on page 56 shows that induced modules are relatively H-free, and the other direction seems like some sort of restatement of Frobenius reciprocity, but since the book doesn't mention the equivalence, I worry it is not true.

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It's important to remember that definition comes with the extra information of the $kH$-submodule with respect to which the module is relatively free. Proposition 8.2 says relatively H-free modules with respect to X exist and Lemma 8.1 says they are unique up to isomorphism. Lemma 8.4 says that the induced module $V^G$ is relatively $H$-free with respect to $V$ and by Lemma 8.1 it is the unique module with this property. Hence every relatively $H$-free $kG$-module is of the form $X^G$ for some $X$, and that $X$ is precisely the submodule with respect to which the module is relatively $H$-free.